# Math Models and Geometry

## Introduction

We are surrounded by all sorts of geometry. Architects use geometry to design buildings. Artists create vivid images out of colorful geometric shapes. Street signs, automobiles, and product packaging all take advantage of geometric properties. In this chapter, we will begin by considering a formal approach to solving problems and use it to solve a variety of common problems, including making decisions about money. Then we will explore geometry and relate it to everyday situations, using the problem-solving strategy we develop.


# Math Models and Geometry

## Use a Problem Solving Strategy

### Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in ?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts like the student in . Read the positive thoughts and say them out loud.
If we take control and believe we can be successful, we will be able to master word problems.
Think of something that you can do now but couldn't do three years ago. Whether it's driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

### Use a Problem-solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.
You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.
Now we'll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We'll demonstrate the strategy as we solve the following problem.
Let's use this approach with another example.
In Solve Sales Tax, Commission, and Discount Applications, we learned how to translate and solve basic percent equations and used them to solve sales tax and commission applications. In the next example, we will apply our Problem Solving Strategy to more applications of percent.

### Solve Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy. Remember to look for clue words such as difference, of, and and.
Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
Consecutive integers are integers that immediately follow each other. Some examples of consecutive integers are:
Notice that each number is one more than the number preceding it. So if we define the first integer as  the next consecutive integer is  The one after that is one more than  so it is  or

### Key Concepts

1. Problem Solving Strategy

### Practice Makes Perfect

Use a Problem-solving Strategy for Word Problems
In the following exercises, use the problem-solving strategy for word problems to solve. Answer in complete sentences.

Solve Number Problems
In the following exercises, solve each number word problem.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Math Models and Geometry

## Solve Money Applications

### Solve Coin Word Problems

Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?
If you can form a step-by-step plan for finding the total value of the coins, it will help you as you begin solving coin word problems.
One way to bring some order to the mess of coins would be to separate the coins into stacks according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, you would add the total value of each pile.
How would you determine the value of each pile? Think about the dime pile—how much is it worth? If you count the number of dimes, you'll know how many you have—the number of dimes.
But this does not tell you the value of all the dimes. Say you counted  dimes, how much are they worth? Each dime is worth —that is the value of one dime. To find the total value of the pile of  dimes, multiply  by  to get  This is the total value of all  dimes.
You could continue this process for each type of coin, and then you would know the total value of each type of coin. To get the total value of all the coins, add the total value of each type of coin.
Let's look at a specific case. Suppose there are  quarters,  dimes,  nickels, and  pennies. We'll make a table to organize the information – the type of coin, the number of each, and the value.
The total value of all the coins is  Notice how  helped us organize all the information. Let's see how this method is used to solve a coin word problem.

### Solve Ticket and Stamp Word Problems

The strategies we used for coin problems can be easily applied to some other kinds of problems too. Problems involving tickets or stamps are very similar to coin problems, for example. Like coins, tickets and stamps have different values; so we can organize the information in tables much like we did for coin problems.
Now we'll do one where we fill in the table all at once.

### Key Concepts

1. Finding the Total Value for Coins of the Same Type

2. Solve a Coin Word Problem
3.

### Practice Makes Perfect

Solve Coin Word Problems
In the following exercises, solve the coin word problems.
Solve Ticket and Stamp Word Problems
In the following exercises, solve the ticket and stamp word problems.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Math Models and Geometry

## Use Properties of Angles, Triangles, and the Pythagorean Theorem

So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems.

### Use the Properties of Angles

Are you familiar with the phrase ‘do a  It means to turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is  degrees. See .
An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In ,  is the angle with vertex at point  The measure of  is written
We measure angles in degrees, and use the symbol  to represent degrees. We use the abbreviation  for the measure of an angle. So if  is  we would write
If the sum of the measures of two angles is  then they are called supplementary angles. In , each pair of angles is supplementary because their measures add to  Each angle is the supplement of the other.
If the sum of the measures of two angles is  then the angles are complementary angles. In , each pair of angles is complementary, because their measures add to  Each angle is the complement of the other.
In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.
The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.
Did you notice that the words complementary and supplementary are in alphabetical order just like  and  are in numerical order?

### Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in  is called  read ‘triangle ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.
The three angles of a triangle are related in a special way. The sum of their measures is

### Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one  angle, which is often marked with the symbol shown in .
If we know that a triangle is a right triangle, we know that one angle measures  so we only need the measure of one of the other angles in order to determine the measure of the third angle.
In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

### Similar Triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles have the same measures.
The two triangles in  are similar. Each side of  is four times the length of the corresponding side of  and their corresponding angles have equal measures.
The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

### Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around  BCE.
Remember that a right triangle has a  angle, which we usually mark with a small square in the corner. The side of the triangle opposite the  angle is called the hypotenuse, and the other two sides are called the legs. See .
The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.
To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation  and defined it in this way:
For example, we found that  is  because
We will use this definition of square roots to solve for the length of a side in a right triangle.

### Key Concepts

1. Supplementary and Complementary Angles
2. Solve Geometry Applications
3. Sum of the Measures of the Angles of a Triangle
4. Right Triangle

5. Properties of Similar Triangles

### Practice Makes Perfect

Use the Properties of Angles
In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle.
In the following exercises, use the properties of angles to solve.
Use the Properties of Triangles
In the following exercises, solve using properties of triangles.
Find the Length of the Missing Side
In the following exercises,  is similar to  Find the length of the indicated side.
On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is  miles.

Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
Find the Length of the Missing Side
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.
In the following exercises, solve. Approximate to the nearest tenth, if necessary.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Math Models and Geometry

## Use Properties of Rectangles, Triangles, and Trapezoids

In this section, we’ll continue working with geometry applications. We will add some more properties of triangles, and we’ll learn about the properties of rectangles and trapezoids.

### Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure (). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure.
When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm) on each side. A square inch is a square that is one inch on each side ().
 shows a rectangular rug that is  feet long by  feet wide. Each square is  foot wide by  foot long, or  square foot. The rug is made of  squares. The area of the rug is square feet.
When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimeters. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimeters, cubic inches, and cubic feet. A cubic centimeter is a cube that measures one centimeter on each side, while a cubic inch is a cube that measures one inch on each side ().
Suppose the cube in  measures  inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find  little cubes, with each one measuring one inch on all sides. So each little cube has a volume of   cubic inch, and the volume of the big cube is  cubic inches.
Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.
Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.
Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.
 shows a square tile that is  inch on each side. If an ant walked around the edge of the tile, it would walk  inches. This distance is the perimeter of the tile.
Since the tile is a square that is  inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.

### Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length,  and the adjacent side as the width,  See .
The perimeter,  of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk  units, or two lengths and two widths. The perimeter then is
What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was  feet long by  feet wide, and its area was  square feet. See . Since  we see that the area,  is the length,  times the width,  so the area of a rectangle is
For easy reference as we work the examples in this section, we will restate the Problem Solving Strategy for Geometry Applications here.
In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

### Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in , we’ve labeled the length  and the width  so it’s area is
We can divide this rectangle into two congruent triangles (). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or  This example helps us see why the formula for the area of a triangle is
The formula for the area of a triangle is  where  is the base and  is the height.
To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a  angle with the base.   shows three triangles with the base and height of each marked.

### Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle.  shows both types of triangles.

### Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base  and the length of the bigger base  The height,  of a trapezoid is the distance between the two bases as shown in .
The formula for the area of a trapezoid is:
Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See .
The height of the trapezoid is also the height of each of the two triangles. See .
The formula for the area of a trapezoid is
If we distribute, we get,

### Key Concepts

1. Properties of Rectangles

2. Triangle Properties

### Practice Makes Perfect

Understand Linear, Square, and Cubic Measure
In the following exercises, determine whether you would measure each item using linear, square, or cubic units.
In the following exercises, find the ⓐ perimeter and ⓑ area of each figure. Assume each side of the square is  cm.
Use the Properties of Rectangles
In the following exercises, find the ⓐ perimeter and ⓑ area of each rectangle.
In the following exercises, solve.
Use the Properties of Triangles
In the following exercises, solve using the properties of triangles.
Use the Properties of Trapezoids
In the following exercises, solve using the properties of trapezoids.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Math Models and Geometry

## Solve Geometry Applications: Circles and Irregular Figures

In this section, we’ll continue working with geometry applications. We will add several new formulas to our collection of formulas. To help you as you do the examples and exercises in this section, we will show the Problem Solving Strategy for Geometry Applications here.
Problem Solving Strategy for Geometry Applications

### Use the Properties of Circles

Do you remember the properties of circles from Decimals and Fractions Together? We’ll show them here again to refer to as we use them to solve applications.
Remember, that we approximate  with  or  depending on whether the radius of the circle is given as a decimal or a fraction. If you use the  key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the  key uses more than two decimal places.
We usually see the formula for circumference in terms of the radius  of the circle:
But since the diameter of a circle is two times the radius, we could write the formula for the circumference in terms
We will use this form of the circumference when we’re given the length of the diameter instead of the radius.

### Find the Area of Irregular Figures

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

### Key Concepts

1. Problem Solving Strategy for Geometry Applications

2. Properties of Circles

### Practice Makes Perfect

Use the Properties of Circles
In the following exercises, solve using the properties of circles.
In the following exercises, find the radius of the circle with given circumference.
Find the Area of Irregular Figures
In the following exercises, find the area of the irregular figure. Round your answers to the nearest hundredth.
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?


# Math Models and Geometry

## Solve Geometry Applications: Volume and Surface Area

In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.
2. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
3. Identify what you are looking for.
4. Name what you are looking for. Choose a variable to represent that quantity.
5. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
6. Solve the equation using good algebra techniques.
7. Check the answer in the problem and make sure it makes sense.
8. Answer the question with a complete sentence.

### Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See ). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.
Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in  has length  units, width  units, and height  units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.
Altogether there are  cubic units. Notice that  is the
The volume,  of any rectangular solid is the product of the length, width, and height.
We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base,  is equal to
We can substitute  for  in the volume formula to get another form of the volume formula.
We now have another version of the volume formula for rectangular solids. Let’s see how this works with the  rectangular solid we started with. See .
To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.
Notice for each of the three faces you see, there is an identical opposite face that does not show.
The surface area  of the rectangular solid shown in  is  square units.
In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its two dimensions, either length and width, length and height, or width and height (see ). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

### Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:
So for a cube, the formulas for volume and surface area are  and

### Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.
Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate  with

### Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height  of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height,  , will be perpendicular to the bases.
Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid,  , can also be used to find the volume of a cylinder.
For the rectangular solid, the area of the base,  , is the area of the rectangular base, length × width. For a cylinder, the area of the base,  is the area of its circular base,   compares how the formula  is used for rectangular solids and cylinders.
To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See .
The distance around the edge of the can is the circumference of the cylinder’s base it is also the length  of the rectangular label. The height of the cylinder is the width  of the rectangular label. So the area of the label can be represented as
To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.
The surface area of a cylinder with radius  and height  is

### Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.
In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See .
Earlier in this section, we saw that the volume of a cylinder is  We can think of a cone as part of a cylinder.  shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.
In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is
Since the base of a cone is a circle, we can substitute the formula of area of a circle,  , for   to get the formula for volume of a cone.
In this book, we will only find the volume of a cone, and not its surface area.

### Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

### Key Concepts

1. Volume and Surface Area of a Rectangular Solid

2. Volume and Surface Area of a Cube

3. Volume and Surface Area of a Sphere

4. Volume and Surface Area of a Cylinder

5. Volume of a Cone

### Practice Makes Perfect

Find Volume and Surface Area of Rectangular Solids
In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions.
In the following exercises, solve.
In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.
In the following exercises, solve.
Find the Volume and Surface Area of Spheres
In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.
Find the Volume and Surface Area of a Cylinder
In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.
Find the Volume of Cones
In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Math Models and Geometry

## Solve a Formula for a Specific Variable

### Use the Distance, Rate, and Time Formula

One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of  miles per hour for  hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said  miles, you already know how to use this formula!
The math to calculate the distance might look like this:
In general, the formula relating distance, rate, and time is
Notice that the units we used above for the rate were miles per hour, which we can write as a ratio  Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.

### Solve a Formula for a Specific Variable

In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft ExcelTM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.
In  and , we used the formula  This formula gives the value of  when you substitute in the values of  and  But in , we had to find the value of  We substituted in values of  and  and then used algebra to solve to  If you had to do this often, you might wonder why there isn’t a formula that gives the value of  when you substitute in the values of  and  We can get a formula like this by solving the formula  for
To solve a formula for a specific variable means to get that variable by itself with a coefficient of  on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.
Let’s try a few examples, starting with the distance, rate, and time formula we used above.
In Solve Simple Interest Applications, we used the formula  to calculate simple interest, where  is interest,  is principal,  is rate as a decimal, and  is time in years.
Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually  and  You might be given an equation that is solved for  and need to solve it for  or vice versa. In the following example, we’re given an equation with both  and  on the same side and we’ll solve it for  To do this, we will follow the same steps that we used to solve a formula for a specific variable.
In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?

### Key Concepts

1. Distance, Rate, and Time

### Section Exercises

### Practice Makes Perfect

Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Solve a Formula for a Specific Variable
In the following exercises, use the formula.
In the following exercises, use the formula
In the following exercises, use the formula
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

### Chapter Review Exercises

### Use a Problem Solving Strategy

Approach Word Problems with a Positive Attitude
In the following exercises, solve.
Use a Problem Solving Strategy for Word Problems
In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Problems
In the following exercises, solve each number word problem.

### Solve Money Applications

Solve Coin Word Problems
In the following exercises, solve each coin word problem.
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.

### Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use Properties of Angles
In the following exercises, solve using properties of angles.
Use Properties of Triangles
In the following exercises, solve using properties of triangles.
In the following exercises,  is similar to  Find the length of the indicated side.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.
In the following exercises, solve. Approximate to the nearest tenth, if necessary.

### Use Properties of Rectangles, Triangles, and Trapezoids

Understand Linear, Square, Cubic Measure
In the following exercises, would you measure each item using linear, square, or cubic measure?
In the following exercises, find
Use Properties of Rectangles
In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle
In the following exercises, solve.
Use Properties of Triangles
In the following exercises, solve using the properties of triangles.
Use Properties of Trapezoids
In the following exercises, solve using the properties of trapezoids.

### Solve Geometry Applications: Circles and Irregular Figures

Use Properties of Circles
In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.
Find the Area of Irregular Figures
In the following exercises, find the area of each shaded region.

### Solve Geometry Applications: Volume and Surface Area

Find Volume and Surface Area of Rectangular Solids
In the following exercises, find the
Find Volume and Surface Area of Spheres
In the following exercises, find the
Find Volume and Surface Area of Cylinders
In the following exercises, find the
Find Volume of Cones
In the following exercises, find the volume of the cone.

### Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula
In the following exercises, solve using the formula for distance, rate, and time.
Solve a Formula for a Specific Variable
In the following exercises, use the formula
In the following exercises, use the formula
In the following exercises, use the formula
In the following exercises, solve.

### Chapter Practice Test


# Polynomials

## Introduction to Polynomials

Expressions known as polynomials are used widely in algebra. Applications of these expressions are essential to many careers, including economists, engineers, and scientists. In this chapter, we will find out what polynomials are and how to manipulate them through basic mathematical operations.


# Polynomials

## Add and Subtract Polynomials

### Identify Polynomials, Monomials, Binomials, and Trinomials

In Evaluate, Simplify, and Translate Expressions, you learned that a term is a constant or the product of a constant and one or more variables. When it is of the form  where  is a constant and  is a whole number, it is called a monomial. A monomial, or a sum and/or difference of monomials, is called a polynomial.
Notice the roots:
1. poly- means many
2. mono- means one
3. bi- means two
4. tri- means three
Here are some examples of polynomials:
Notice that every monomial, binomial, and trinomial is also a polynomial. They are special members of the family of polynomials and so they have special names. We use the words ‘monomial’, ‘binomial’, and  ‘trinomial’ when referring to these special polynomials and just call all the rest ‘polynomials’.

### Determine the Degree of Polynomials

In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.
A monomial that has no variable, just a constant, is a special case. The degree of a constant is —it has no variable.
Let's see how this works by looking at several polynomials. We'll take it step by step, starting with monomials, and then progressing to polynomials with more terms.
Remember: Any base written without an exponent has an implied exponent of
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form. Look back at the polynomials in . Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.

### Add and Subtract Monomials

In The Language of Algebra, you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

### Add and Subtract Polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.
Parentheses are grouping symbols. When we add polynomials as we did in , we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

### Evaluate a Polynomial for a Given Value

In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify.

### Practice Makes Perfect

Identify Polynomials, Monomials, Binomials and Trinomials
In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value
In the following exercises, evaluate each polynomial for the given value.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Polynomials

## Use Multiplication Properties of Exponents

### Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example,  means to multiply four factors of  so  means  This format is known as exponential notation.
In the expression  the exponent tells us how many times we use the base  as a factor.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

### Simplify Expressions Using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.
First, we will look at an example that leads to the Product Property.
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

An example with numbers helps to verify this property.
We can extend the Product Property of Exponents to more than two factors.

### Simplify Expressions Using the Power Property of Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.
We multiplied the exponents. This leads to the Power Property for Exponents.

An example with numbers helps to verify this property.

### Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.
The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

### Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

### Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

### Key Concepts

1. Exponential Notation

This is read

2. Product Property of Exponents

3. Power Property for Exponents

4. Product to a Power Property for Exponents

### Practice Makes Perfect

Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
Simplify Expressions Using the Product Property of Exponents
In the following exercises, simplify each expression using the Product Property of Exponents.
Simplify Expressions Using the Power Property of Exponents
In the following exercises, simplify each expression using the Power Property of Exponents.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises, multiply the following monomials.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Polynomials

## Multiply Polynomials

### Multiply a Polynomial by a Monomial

In Distributive Property you learned to use the Distributive Property to simplify expressions such as  You multiplied both terms in the parentheses,  by  to get  With this chapter's new vocabulary, you can say you were multiplying a binomial,  by a monomial,  Multiplying a binomial by a monomial is nothing new for you!
Multiplying a monomial by a trinomial works in much the same way.
Now we will have the monomial as the second factor.

### Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

### Using the Distributive Property

We will start by using the Distributive Property. Look again at .
Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.
Be careful to distinguish between a sum and a product.
Now we'll see how to multiply binomials where the variable has a coefficient.
In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.
Up to this point, the product of two binomials has been a trinomial. This is not always the case.

### Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.
Where did the first term,  come from?
It is the product of  the first terms in
The next term,  is the product of  the two outer terms.
The third term,  is the product of  the two inner terms.
And the last term,  came from multiplying the two last terms.
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.
Let's look at  again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

### Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
You start by multiplying  by  to get
Then you multiply  by  lining up the partial product in the correct columns.
Last, you add the partial products.
Now we'll apply this same method to multiply two binomials.
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.

### Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.
Now let's do this same multiplication using the Vertical Method.

### Key Concepts

1. Use the FOIL method for multiplying two binomials.

2. Multiplying Two Binomials: To multiply binomials, use the:

3. Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:

### Practice Makes Perfect

Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method

In the following exercises, multiply the following binomials. Use any method.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Polynomials

## Divide Monomials

### Simplify Expressions Using the Quotient Property of Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients.
As before, we'll try to discover a property by looking at some examples.

Notice that in each case the bases were the same and we subtracted the exponents.
1. When the larger exponent was in the numerator, we were left with factors in the numerator and  in the denominator, which we simplified.
2. When the larger exponent was in the denominator, we were left with factors in the denominator, and  in the numerator, which could not be simplified.
We write:
A couple of examples with numbers may help to verify this property.
When we work with numbers and the exponent is less than or equal to  we will apply the exponent. When the exponent is greater than , we leave the answer in exponential form.

### Simplify Expressions with Zero Exponents

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like  From earlier work with fractions, we know that
In words, a number divided by itself is  So  for any  (), since any number divided by itself is
The Quotient Property of Exponents shows us how to simplify  when  and when  by subtracting exponents. What if ?
Now we will simplify  in two ways to lead us to the definition of the zero exponent.
Consider first  which we know is

We see  simplifies to a  and to . So .
In this text, we assume any variable that we raise to the zero power is not zero.
Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.
What about raising an expression to the zero power? Let's look at  We can use the product to a power rule to rewrite this expression.

This tells us that any non-zero expression raised to the zero power is one.

### Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.
Notice that the exponent applies to both the numerator and the denominator.
We see that  is

This leads to the Quotient to a Power Property for Exponents.

An example with numbers may help you understand this property:

### Simplify Expressions by Applying Several Properties

We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

### Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.
When we divide monomials with more than one variable, we write one fraction for each variable.
Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.
In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction.

### Key Concepts

1. Equivalent Fractions Property

2. Zero Exponent

3. Quotient Property for Exponents

4. Quotient to a Power Property for Exponents

### Practice Makes Perfect

Simplify Expressions Using the Quotient Property of Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.

### Mixed Practice

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Polynomials

## Integer Exponents and Scientific Notation

### Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.
What if we just subtract exponents, regardless of which is larger? Let’s consider
We subtract the exponent in the denominator from the exponent in the numerator.
We can also simplify  by dividing out common factors:
This implies that  and it leads us to the definition of a negative exponent.
The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.
We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.
Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form,  where  and m and n are integers.
When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents,

### Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.
If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
To simplify a fraction, we use the Quotient Property.

### Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of  We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.
Consider the numbers  and  We know that  means  and  means  If we write the  as a power of ten in exponential form, we can rewrite these numbers in this way:
When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than  and the second factor is a power of  written in exponential form, it is said to be in scientific notation.
It is customary in scientific notation to use  as the multiplication sign, even though we avoid using this sign elsewhere in algebra.
Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.
If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.
In both cases, the decimal was moved  places to get the first factor,  by itself.
1. The power of  is positive when the number is larger than
2. The power of  is negative when the number is between  and

### Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.
If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.
In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

### Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

### Key Concepts

1. Summary of Exponent Properties

2. Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:

3. Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:

### Practice Makes Perfect

Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply or divide and write your answer in decimal form.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?


# Polynomials

## Introduction to Factoring Polynomials

### Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.
In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
First we will find the greatest common factor of two numbers.
In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.
In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

### Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example,  as  in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:
Here, we will start with a product, like  and end with its factors,  To do this we apply the Distributive Property “in reverse”.
The form on the left is used to multiply. The form on the right is used to factor.
So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!
Notice that in , we used the word factor as both a noun and a verb:
The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
In the next example, we factor a variable from a binomial.
When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!
When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.
Pay close attention to the signs of the terms in the next example.

### Key Concepts

1. Find the greatest common factor.

2. Distributive Property

3. Factor the greatest common factor from a polynomial.

### Section Exercises

### Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

### Chapter Review Exercises

### Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value of the Variable
In the following exercises, evaluate each polynomial for the given value.

### Use Multiplication Properties of Exponents

Simplify Expressions with Exponents
In the following exercises, simplify.
Simplify Expressions Using the Product Property of Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Power Property of Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises, multiply the monomials.

### Multiply Polynomials

Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using various methods.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using any method.

### Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.

### Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply and write your answer in decimal form.

### Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.

### Chapter Practice Test

In the following exercises, simplify each expression.
In the following exercises, factor the greatest common factor from each polynomial.
In the following exercises, simplify, and write your answer in decimal form.


# Graphs

## Graphs

Which cyclist will win the race? What will the winning time be? How many seconds will separate the winner from the runner-up? One way to summarize the information from the race is by creating a graph. In this chapter, we will discuss the basic concepts of graphing. The applications of graphing go far beyond races. They are used to present information in almost every field, including healthcare, business, and entertainment.


# Graphs

## Use the Rectangular Coordinate System

### Plot Points on a Rectangular Coordinate System

Many maps, such as the Campus Map shown in , use a grid system to identify locations. Do you see the numbers  and  across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.
For example, the Student Center is in section 2B. It is located in the grid section above the number  and next to the letter B. In which grid section is the Stadium? The Stadium is in section 4D.
Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. To create a rectangular coordinate system, start with a horizontal number line. Show both positive and negative numbers as you did before, using a convenient scale unit. This horizontal number line is called the .
Now, make a vertical number line passing through the  at  Put the positive numbers above  and the negative numbers below  See . This vertical line is called the .
Vertical grid lines pass through the integers marked on the  Horizontal grid lines pass through the integers marked on the  The resulting grid is the rectangular coordinate system.
The rectangular coordinate system is also called the  plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named René Descartes.)
The  and the  form the rectangular coordinate system.  These axes divide a plane into four areas, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See .
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.
So how do the coordinates of a point help you locate a point on the  plane?
Let’s try locating the point . In this ordered pair, the -coordinate is  and the -coordinate is .
We start by locating the  value,  on the  Then we lightly sketch a vertical line through  as shown in .
Now we locate the  value,  on the -axis and sketch a horizontal line through . The point where these two lines meet is the point with coordinates  We plot the point there, as shown in .
How do the signs affect the location of the points?
You may have noticed some patterns as you graphed the points in the two previous examples.
For each point in Quadrant IV, what do you notice about the signs of the coordinates?
What about the signs of the coordinates of the points in the third quadrant? The second quadrant? The first quadrant?
Can you tell just by looking at the coordinates in which quadrant the point (−2, 5) is located? In which quadrant is (2, −5) located?

We can summarize sign patterns of the quadrants as follows. Also see .
What if one coordinate is zero? Where is the point  located? Where is the point  located? The point   is on the y-axis and the point   is on the x-axis.

### Identify Points on a Graph

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the  of a point on a graph, read the number on the  directly above or below the point. To identify the  of a point, read the number on the  directly to the left or right of the point. Remember, to write the ordered pair using the correct order

### Verify Solutions to an Equation in Two Variables

All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as  Then we checked the solution by substituting back into the equation.
Here’s an example of a linear equation in one variable, and its one solution.
But equations can have more than one variable. Equations with two variables can be written in the general form  An equation of this form is called a linear equation in two variables.
Notice that the word “line” is in linear.
Here is an example of a linear equation in two variables,  and
Is  a linear equation? It does not appear to be in the form  But we could rewrite it in this form.
By rewriting  as  we can see that it is a linear equation in two variables because it can be written in the form
Linear equations in two variables have infinitely many solutions. For every number that is substituted for  there is a corresponding  value. This pair of values is a solution to the linear equation and is represented by the ordered pair  When we substitute these values of  and  into the equation, the result is a true statement because the value on the left side is equal to the value on the right side.

### Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the  of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for  and then solve the equation for  Or, choose a value for  and then solve for
We’ll start by looking at the solutions to the equation  we found in . We can summarize this information in a table of solutions.
To find a third solution, we’ll let  and solve for
The ordered pair is a solution to .  We will add it to the table.

We can find more solutions to the equation by substituting any value of  or any value of  and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.

### Find Solutions to Linear Equations in Two Variables

To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either  or  We could choose  or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose  as one of our values.
We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation
Let’s find some solutions to another equation now.

### Key Concepts

1. Sign Patterns of the Quadrants

2. Coordinates of Zero

### Practice Makes Perfect

Plot Points on a Rectangular Coordinate System
In the following exercises, plot each point on a coordinate grid.
In the following exercises, plot each point on a coordinate grid and identify the quadrant in which the point is located.
In the following exercises, plot each point on a coordinate grid.
Identify Points on a Graph
In the following exercises, name the ordered pair of each point shown.
Verify Solutions to an Equation in Two Variables
In the following exercises, determine which ordered pairs are solutions to the given equation.
Find Solutions to Linear Equations in Two Variables
In the following exercises, complete the table to find solutions to each linear equation.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Graphs

## Graphing Linear Equations

### Recognize the Relation Between the Solutions of an Equation and its Graph

In Use the Rectangular Coordinate System, we found a few solutions to the equation . They are listed in the table below.  So, the ordered pairs , , , , are some solutions to the equation. We can plot these solutions in the rectangular coordinate system as shown on the graph at right.

Notice how the points line up perfectly?  We connect the points with a straight line to get the graph of the equation .  Notice the arrows on the ends of each side of the line.  These arrows indicate the line continues.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions!
Notice that the point whose coordinates are   is on the line shown in . If you substitute   and   into the equation, you find that it is a solution to the equation.

So   is not a solution to the equation  . Therefore the point  is not on the line.

This is an example of the saying,” A picture is worth a thousand words.”  The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation .

### Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method.
Let’s graph the equation  by plotting points.
We start by finding three points that are solutions to the equation. We can choose any value for  or  and then solve for the other variable.
Since  is isolated on the left side of the equation, it is easier to choose values for  We will use  and  for  for this example. We substitute each value of  into the equation and solve for
We can organize the solutions in a table. See .
Now we plot the points on a rectangular coordinate system. Check that the points line up. If they did not line up, it would mean we made a mistake and should double-check all our work. See .
Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. The line is the graph of
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you plot only two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line. If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. See .
When an equation includes a fraction as the coefficient of  we can substitute any numbers for  But the math is easier if we make ‘good’ choices for the values of  This way we will avoid fraction answers, which are hard to graph precisely.
So far, all the equations we graphed had  given in terms of  Now we’ll graph an equation with  and  on the same side.
In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation  If  is  what is the value of
The solution is the point  This point has a fraction for the -coordinate. While we could graph this point, it is hard to be precise graphing fractions. Remember in the example  we carefully chose values for  so as not to graph fractions at all. If we solve the equation  for  it will be easier to find three solutions to the equation.
Now we can choose values for  that will give coordinates that are integers. The solutions for  and  are shown.

### Graph Vertical and Horizontal Lines

Can we graph an equation with only one variable? Just  and no  or just  without an  How will we make a table of values to get the points to plot?
Let’s consider the equation  The equation says that  is always equal to  so its value does not depend on  No matter what  is, the value of  is always
To make a table of solutions, we write  for all the  values. Then choose any values for  Since  does not depend on  you can chose any numbers you like. But to fit the size of our coordinate graph, we’ll use  and  for the -coordinates as shown in the table.
Then plot the points and connect them with a straight line. Notice in  that the graph is a vertical line.
What if the equation has  but no ? Let’s graph the equation  This time the -value is a constant, so in this equation  does not depend on
To make a table of solutions, write  for all the  values and then choose any values for
We’ll use  and  for the -values.
Plot the points and connect them, as shown in . This graph is a horizontal line passing through the  at
The equations for vertical and horizontal lines look very similar to equations like  What is the difference between the equations  and
The equation  has both  and  The value of  depends on the value of  The  changes according to the value of
The equation  has only one variable. The value of  is constant. The  is always  It does not depend on the value of
The graph shows both equations.
Notice that the equation  gives a slanted line whereas  gives a horizontal line.

### Key Concepts

1. Graph a linear equation by plotting points.
2. Graph of a Linear Equation:The graph of a linear equation  is a straight line.

### Practice Makes Perfect

Recognize the Relation Between the Solutions of an Equation and its Graph
In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide

Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph the vertical and horizontal lines.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
Mixed Practice

In the following exercises, graph each equation.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Graphs

## Graphing with Intercepts

### Identify the Intercepts on a Graph

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.
At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line.
Let’s look at the graph of the lines shown in .
First, notice where each of these lines crosses the x- axis:
Do you see a pattern?
For each row, the y- coordinate of the point where the line crosses the x- axis is zero.  The point where the line crosses the x- axis has the form ; and is called the x-intercept of the line.  The x- intercept occurs when y is zero.
Now, let's look at the points where these lines cross the y-axis.

### Find the Intercepts from an Equation of a Line

Recognizing that the  occurs when  is zero and that the  occurs when  is zero gives us a method to find the intercepts of a line from its equation. To find the  let  and solve for  To find the  let  and solve for

### Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

### Choose the Most Convenient Method to Graph a Line

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.
What is it about the form of equation that can help us choose the most convenient method to graph its line?
Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.
Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.
In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form . We substituted  and  to find the x- and y- intercepts, and then found a third point by choosing a value for x or y.
This leads to the following strategy for choosing the most convenient method to graph a line.

### Key Concepts

1. Intercepts
2. Find the
3. Graph a line using the intercepts
4. Choose the most convenient method to graph a line

### Practice Makes Perfect

Identify the Intercepts on a Graph
In the following exercises, find the  and  intercepts.
Find the
In the following exercises, find the intercepts.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
Choose the Most Convenient Method to Graph a Line
In the following exercises, identify the most convenient method to graph each line.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Graphs

## Understand Slope of a Line

As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat?
The steepness of the slant of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

### Use Geoboards to Model Slope

In this section, we will explore the concepts of slope.
Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
We’ll start by stretching a rubber band between two pegs to make a line as shown in .
Does it look like a line?
Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in . We carefully make a  angle around the third peg, so that one side is vertical and the other is horizontal.
To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the  and the horizontal distance is called the , as shown in .
To help remember the terms, it may help to think of the images shown in .

On our geoboard, the rise is  units because the rubber band goes up  spaces on the vertical leg. See .
What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across  spaces on the horizontal leg, so the run is  units.
The slope of a line is the ratio of the rise to the run. So the slope of our line is  In mathematics, the slope is always represented by the letter
What is the slope of the line on the geoboard in ?
When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.
If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.
Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.
Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in .

As you read from left to right, the line in Figure A, is going up; it has positive slope.  The line Figure B is going down; it has negative slope.

### Find the Slope of a Line from its Graph

Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.
To find the slope, we must count out the rise and the run. But where do we start?
We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
The lines in the previous examples had -intercepts with integer values, so it was convenient to use the  as one of the points we used to find the slope. In the next example, the -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.

### Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.
1. horizontal line  all the -coordinates are the same.
2. vertical line  all the -coordinates are the same.
So how do we find the slope of the horizontal line  One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens in . We’ll use the two points  and  to count the rise and run.
The slope of the horizontal line  is
All horizontal lines have slope . When the -coordinates are the same, the rise is .
Now we’ll consider a vertical line, such as the line , shown in . We’ll use the two points  and  to count the rise and run.
But we can’t divide by  Division by  is undefined. So we say that the slope of the vertical line  is undefined. The slope of all vertical lines is undefined, because the run is

### Use the Slope Formula to find the Slope of a Line between Two Points

Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.
Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair  gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol  be used to represent two different points?
Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.
1.
2.
We will use  to identify the first point and  to identify the second point. If we had more than two points, we could use  and so on.
To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points  and  in .
Since we have two points, we will use subscript notation.
On the graph, we counted the rise of  The rise can also be found by subtracting the  of the points.
We counted a run of  The run can also be found by subtracting the
We’ve shown that  is really another version of  We can use this formula to find the slope of a line when we have two points on the line.
Say the formula to yourself to help you remember it:
Doing the Manipulative Mathematics activity  “Slope of Lines Between Two Points” will help you develop a better understanding of how to find the slope of a line between two points.
How do we know which point to call #1 and which to call #2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.
The slope is the same no matter which order we use the points.

### Graph a Line Given a Point and the Slope

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.
Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.

### Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.
Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.

### Key Concepts

1. Find the slope from a graph
2. Slope of a Horizontal Line
3. Slope of a Vertical Line
4. Slope Formula
5. Graph a line given a point and a slope.

### Section Exercises

### Practice Makes Perfect

Use Geoboards to Model Slope
In the following exercises, find the slope modeled on each geoboard.
In the following exercises, model each slope. Draw a picture to show your results.
Find the Slope of a Line from its Graph
In the following exercises, find the slope of each line shown.
Find the Slope of Horizontal and Vertical Lines
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph the line given a point and the slope.
Solve Slope Applications
In the following exercises, solve these slope applications.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises

### Use the Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
Identify Points on a Graph
In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.
Verify Solutions to an Equation in Two Variables
In the following exercises, find the ordered pairs that are solutions to the given equation.
Complete a Table of Solutions to a Linear Equation in Two Variables
In the following exercises, complete the table to find solutions to each linear equation.
Find Solutions to a Linear Equation in Two Variables
In the following exercises, find three solutions to each linear equation.

### Graphing Linear Equations

Recognize the Relation Between the Solutions of an Equation and its Graph
In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide

Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph the vertical or horizontal lines.

### Graphing with Intercepts

Identify the Intercepts on a Graph
In the following exercises, find the  and
Find the Intercepts from an Equation of a Line
In the following exercises, find the intercepts.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
Choose the Most Convenient Method to Graph a Line
In the following exercises, identify the most convenient method to graph each line.

### Understand Slope of a Line

Use Geoboards to Model Slope
In the following exercises, find the slope modeled on each geoboard.
In the following exercises, model each slope. Draw a picture to show your results.
Find the Slope of a Line from its Graph
In the following exercises, find the slope of each line shown.
Find the Slope of Horizontal and Vertical Lines
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph the line given a point and the slope.
Solve Slope Applications
In the following exercise, solve the slope application.

### Chapter Practice Test

In the following exercises, find three solutions to each equation and then graph each line.
In the following exercises, find the slope of each line.