# Math Models

## Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

### Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.
We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.
The plural of the word vertex is vertices. All triangles have three vertices. Triangles are named by their vertices: The triangle in  is called
The three angles of a triangle are related in a special way. The sum of their measures is  Note that we read  as “the measure of angle A.” So in  in ,
Because the perimeter of a figure is the length of its boundary, the perimeter of  is the sum of the lengths of its three sides.
To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a  angle with the base. We will draw  again, and now show the height, h. See .

The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one  angle, which we usually mark with a small square in the corner.
In the examples we have seen 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. We will wait to draw the figure until we write expressions for all the angles we are looking for.

### Use the Pythagorean Theorem

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem. This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.
Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a  angle, marked with a small square in the corner. The side of the triangle opposite the  angle is called the hypotenuse and each of the other sides are called legs.
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 lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle,  where  are the lengths of the legs and  is the length of the hypotenuse.
Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.
To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation  and the definition:
If  then  for
For example, we found that  is 5 because
Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

### Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have 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, L, and its adjacent side as the width, W.
The distance around this rectangle is  or  This is the perimeter, P, of the rectangle.
What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.
The area is the length times the width.
The formula for the area of a rectangle is

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

### Key Concepts

1. Problem-Solving Strategy for Geometry Applications

2. Triangle Properties For
Angle measures:

Perimeter:

Area:

A right triangle has one  angle.
3. The Pythagorean Theorem In any right triangle,  where c is the length of the hypotenuse and a and b are the lengths of the legs.
4. Properties of Rectangles

### Practice Makes Perfect

Solving Applications Using Triangle Properties
In the following exercises, solve using triangle properties.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth, if necessary.
In the following exercises, solve using the Pythagorean Theorem. Approximate to the nearest tenth, if necessary.
Solve Applications Using Rectangle Properties
In the following exercises, solve using rectangle properties.

### 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

## Solve Uniform Motion Applications

### Solve Uniform Motion Applications

When planning a road trip, it often helps to know how long it will take to reach the destination or how far to travel each day. We would use the distance, rate, and time formula,  which we have already seen.
In this section, we will use this formula in situations that require a little more algebra to solve than the ones we saw earlier. Generally, we will be looking at comparing two scenarios, such as two vehicles travelling at different rates or in opposite directions. When the speed of each vehicle is constant, we call applications like this uniform motion problems.
Our problem-solving strategies will still apply here, but we will add to the first step. The first step will include drawing a diagram that shows what is happening in the example. Drawing the diagram helps us understand what is happening so that we will write an appropriate equation. Then we will make a table to organize the information, like we did for the money applications.
The steps are listed here for easy reference:
In , the last example, we had two trains traveling the same distance. The diagram and the chart helped us write the equation we solved. Let’s see how this works in another case.
As you read the next example, think about the relationship of the distances traveled. Which of the previous two examples is more similar to this situation?
In the distance, rate, and time formula, time represents the actual amount of elapsed time (in hours, minutes, etc.). If a problem gives us starting and ending times as clock times, we must find the elapsed time in order to use the formula.

### Key Concepts

1. Distance, Rate, and Time

2. Problem-Solving Strategy—Distance, Rate, and Time Applications

### Practice Makes Perfect

Solve Uniform Motion Applications
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.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Math Models

## Solve Applications with Linear Inequalities

### Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for $20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return? How much money should each family member’s holiday gift cost without going over budget?
The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations. We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.
Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down. To check applications like this, we will round our answer to a number that is easy to compute with and make sure that number makes the inequality true.
A common goal of most businesses is to make a profit. Profit is the money that remains when the expenses have been subtracted from the money earned. In the next example, we will find the number of jobs a small businessman needs to do every month in order to make a certain amount of profit.
Sometimes life gets complicated! There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.

### Key Concepts

1. Solving inequalities

### Section Exercises

### Practice Makes Perfect

Solve Applications with Linear Inequalities
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.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter 3 Review Exercises

### 3.1 Using a Problem Solving Strategy

Approach Word Problems with a Positive Attitude
In the following exercises, reflect on your approach to word problems.
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.

### 3.2 Solve Percent Applications

Translate and Solve Basic Percent Equations
In the following exercises, translate and solve.
Solve Percent Applications
In the following exercises, solve.
Find Percent Increase and Percent Decrease
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.
Solve Applications with Discount or Mark-up
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

### 3.3 Solve Mixture 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.
Solve Mixture Word Problems
In the following exercises, solve each mixture word problem.

### 3.4 Solve Geometry Applications: Triangles, Rectangles and the Pythagorean Theorem

Solve Applications Using Triangle Properties
In the following exercises, solve using triangle properties.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
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.
Solve Applications Using Rectangle Properties
In the following exercises, solve using rectangle properties.

### 3.5 Solve Uniform Motion Applications

Solve Uniform Motion Applications
In the following exercises, solve.

### 3.6 Solve Applications with Linear Inequalities

Solve Applications with Linear Inequalities
In the following exercises, solve.

### Practice Test

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.


# Graphs

## Introduction

Graphs are found in all areas of our lives—from commercials showing you which cell phone carrier provides the best coverage, to bank statements and news articles, to the boardroom of major corporations. In this chapter, we will study the rectangular coordinate system, which is the basis for most consumer graphs. We will look at linear graphs, slopes of lines, equations of lines, and linear inequalities.


# Graphs

## Use the Rectangular Coordinate System

### Plot Points on a Rectangular Coordinate System

Just like 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. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.
The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, 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  of the point, and the second number is the  of the point.
The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is . The point  has a special name. It is called the origin.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point  as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through . Then, locate 3 on the y-axis and sketch a horizontal line through . Now, find the point where these two lines meet—that is the point with coordinates .
Notice that the vertical line through  and the horizontal line through  are not part of the graph. We just used them to help us locate the point .
How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.
For the point in  in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of 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  is located? In which quadrant is  located?
What if one coordinate is zero as shown in ? 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.
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 x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, when you write the ordered pair use the correct order, .

### Verify Solutions to an Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like . (Then, you checked the solution by substituting back into the equation.)
Here’s an example of an equation in one variable, and its one solution.
But equations can have more than one variable. Equations with two variables may be of the form . Equations of this form are called linear equations in two variables.
Notice the word line in linear. Here is an example of a linear equation in two variables,  and .
The equation  is also a linear equation. But it does not appear to be in the form . We can use the Addition Property of Equality and rewrite it in  form.
By rewriting  as , we can easily see that it is a linear equation in two variables because it is of the form . When an equation is in the form , we say it is in standard form.
Most people prefer to have , , and  be integers and  when writing a linear equation in standard form, although it is not strictly necessary.
Linear equations 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 Two Variables

In the examples above, we substituted the x- and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for  and then solve the equation for . Or, pick a value for  and then solve for .
We’ll start by looking at the solutions to the equation  that we found in . We can summarize this information in a table of solutions, as shown in .
To find a third solution, we’ll let  and solve for .
The ordered pair  is a solution to . We will add it to .
We can find more solutions to the equation by substituting in any value of  or any value of  and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

### Find Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for  or  But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.
When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of  and then solve for .
We have seen how using zero as one value of  makes finding the value of  easy. When an equation is in standard form, with both the  and  on the same side of the equation, it is usually easier to first find one solution when  find a second solution when , and then find a third solution.

### Key Concepts

1. Sign Patterns of the Quadrants

2. Points on the Axes

3. Solution of a Linear Equation

### Practice Makes Perfect

Plot Points 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.
In the following exercises, plot each point in a rectangular coordinate system.
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, which ordered pairs are solutions to the given equations?
Complete a Table of Solutions to a Linear Equation
In the following exercises, complete the table to find solutions to each linear equation.
Find Solutions to a Linear Equation
In the following exercises, find three 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

## Graph  Linear Equations in Two Variables

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

In the previous section, we found several solutions to the equation . They are listed in . So, the ordered pairs , , and  are some solutions to the equation . We can plot these solutions in the rectangular coordinate system as shown in .
Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation . See . 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 the point  is a solution to the equation . (The phrase “the point whose coordinates are ” is often shortened to “the point .”)
So  is not a solution to the equation . Therefore, the point  is not on the line. See . 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 to graph  is called plotting points, or the Point–Plotting Method.
The steps to take when graphing a linear equation by plotting points are summarized below.
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 only plot 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. Look at the difference between part (a) and part (b) in .
Let’s do another example. This time, we’ll show the last two steps all on one grid.
When an equation includes a fraction as the coefficient of , we can still 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. Let’s see what happens in the equation . If  what is the value of ?
This point has a fraction for the x- coordinate and, 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.
The solutions for , , and  are shown in the . The graph is shown in .
Can you locate the point , which we found by letting , on the line?
If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x- and y-axis are the same, the graphs match!
The equation in  was written in standard form, with both  and  on the same side. We solved that equation for  in just one step. But for other equations in standard form it is not that easy to solve for , so we will leave them in standard form. We can still find a first point to plot by letting  and solving for . We can plot a second point by letting  and then solving for . Then we will plot a third point by using some other value for  or .

### 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 . This equation has only one variable, . The equation says that  is always equal to , so its value does not depend on . No matter what  is, the value of  is always .
So to make a table of values, write  in for all the  values. Then choose any values for . Since  does not depend on , you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See .
Plot the points from  and connect them with a straight line. Notice in  that we have graphed a vertical line.

What if the equation has  but no ? Let’s graph the equation . This time the y- value is a constant, so in this equation,  does not depend on . Fill in 4 for all the ’s in  and then choose any values for . We’ll use 0, 2, and 4 for the x-coordinates.
The graph is a horizontal line passing through the y-axis at 4. See .

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 y-coordinate changes according to the value of . The equation  has only one variable. The value of  is constant. The y-coordinate is always 4. It does not depend on the value of . See .
Notice, in , the equation  gives a slanted line, while  gives a horizontal line.

### Key Concepts

1. Graph a Linear Equation by Plotting Points

### Practice Makes Perfect

Recognize the Relationship Between the Solutions of an Equation and its Graph
In the following exercises, for each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation? ⓑ Is the point on the line?
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 each equation.
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 goals?


# Graphs

## Graph with Intercepts

### Identify the x- and y- Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.
At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.
Let’s look at the graphs of the lines in .
First, notice where each of these lines crosses the  negative axis. See .
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 . The x- intercept occurs when  is zero.
Now, let’s look at the points where these lines cross the y- axis. See .
What is the pattern here?
In each row, the x- coordinate of the point where the line crosses the y- axis is zero. The point where the line crosses the y- axis has the form  and is called the y- intercept of the line. The y- intercept occurs when  is zero.

### Find the x- and y- Intercepts from an Equation of a Line

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

### Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.
The steps to graph a linear equation using the intercepts are summarized below.

### Key Concepts

1. Find the

2. Graph a Linear Equation using the Intercepts

3. Strategy for Choosing the Most Convenient Method to Graph a Line:

### Practice Makes Perfect

Identify the
In the following exercises, find the x- and y- intercepts on each graph.
Find the
In the following exercises, find the intercepts for each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.

### 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

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?
In mathematics, the ‘tilt’ 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, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
In this section, we will explore the concept of slope.

### Use Geoboards to Model Slope

A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us 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.
We’ll start by stretching a rubber band between two pegs as shown in .
Doesn’t 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 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in .
If our geoboard and rubber band look just like the one shown in , the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)
What is the run?
The rubber band goes across 3 units. The run is 3 (see ).
The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter .
What is the slope of the line on the geoboard in ?
The line has slope . This means that the line rises 2 units for every 3 units of run.
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. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.
Notice that in  the slope is positive and in  the slope is negative. Do you notice any difference in the two lines shown in (a) and (b)?
We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in (a) is going up; it has positive slope. The line in (b) is going down; it has negative slope.

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

Now, we’ll look at some graphs on the -coordinate plane and see how 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 two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.

### 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.
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 when we do this.
All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.
The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.
Now, we’ll consider a vertical line, the line.
But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line  is undefined.
The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.
Remember, we ‘read’ a line from left to right, just like we read written words in English.

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

Sometimes we’ll need to find the slope of a line between two points when we don’t 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 as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some 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 the points.
The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?
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.
Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points  and .
Since we have two points, we will use subscript notation, .
On the graph, we counted the rise of 3 and the run of 5.
Notice that the rise of 3 can be found by subtracting the y-coordinates 6 and 3.
And the run of 5 can be found by subtracting the x-coordinates 7 and 2.
We know . So .
We rewrite the rise and run by putting in the coordinates .
But 6 is , the y-coordinate of the second point and 3 is , the y-coordinate of the first point.
So we can rewrite the slope using subscript notation.
Also, 7 is , the x-coordinate of the second point and 2 is , the x-coordinate of the first point.
So, again, we rewrite the slope using subscript notation.
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.

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

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.
One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then 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.

### Key Concepts

1. Find the Slope of a Line from its Graph using

2. Graph a Line Given a Point and the Slope

3. Slope of a Horizontal Line

4. Slope of a vertical line

### 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.
Use  to 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 each line with the given point and slope.

### 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?


# Graphs

## Use the Slope-Intercept Form of an Equation of a Line

### Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.
In Graph Linear Equations in Two Variables, we graphed the line of the equation  by plotting points. See . Let’s find the slope of this line.
The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula:
What is the y-intercept of the line? The y-intercept is where the line crosses the y-axis, so y-intercept is . The equation of this line is:
Notice, the line has:
When a linear equation is solved for , the coefficient of the  term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation  is in slope–intercept form.
Sometimes the slope–intercept form is called the “y-form.”

### Identify the Slope and y-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y-intercept from the equation of a line.
When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for .

### Graph a Line Using its Slope and Intercept

Now that we know how to find the slope and y-intercept of a line from its equation, we can graph the line by plotting the y-intercept and then using the slope to find another point.
We have used a grid with  and  both going from about  to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.
Now that we have graphed lines by using the slope and y-intercept, let’s summarize all the methods we have used to graph lines. See .

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

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?
While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.
Here are six equations we graphed in this chapter, and the method we used to graph each of them.
Equations #1 and #2 each have just one variable. Remember, in equations of this form 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 #3 and #4, both  and  are on the same side of the equation. These two equations are of the form . We substituted  to find the x-intercept and  to find the y-intercept, and then found a third point by choosing another value for  or .
Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.
This leads to the following strategy.

### Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.
Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of x and y. The variable names remind us of what quantities are being measured.
The cost of running some types business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

### Use Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.
We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See .
What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different x-intercepts are parallel. See .
Let’s graph the equations  and  on the same grid. The first equation is already in slope–intercept form: . We solve the second equation for :
Graph the lines.
Notice the lines look parallel. What is the slope of each line? What is the y-intercept of each line?
The slopes of the lines are the same and the y-intercept of each line is different. So we know these lines are parallel.
Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.

### Use Slopes to Identify Perpendicular Lines

Let’s look at the lines whose equations are  and , shown in .
These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.
What do you notice about the slopes of these two lines? As we read from left to right, the line  rises, so its slope is positive. The line drops from left to right, so it has a negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?
If we look at the slope of the first line, , and the slope of the second line, , we can see that they are negative reciprocals of each other. If we multiply them, their product is
This is always true for perpendicular lines and leads us to this definition.
We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.
We find the slope–intercept form of the equation, and then see if the slopes are negative reciprocals. If the product of the slopes is , the lines are perpendicular. Perpendicular lines may have the same y-intercepts.

### Key Concepts

1. The slope–intercept form of an equation of a line with slope  and y-intercept,  is, .
2. Graph a Line Using its Slope and

3. Strategy for Choosing the Most Convenient Method to Graph a Line: Consider the form of the equation.

4. Parallel lines are lines in the same plane that do not intersect.

5. Perpendicular lines are lines in the same plane that form a right angle.

### Practice Makes Perfect

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line
In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation .
Identify the Slope and y-Intercept From an Equation of a Line
In the following exercises, identify the slope and y-intercept of each line.
Graph a Line Using Its Slope and Intercept
In the following exercises, graph the line of each equation using its slope and y-intercept.
Choose the Most Convenient Method to Graph a Line
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope–Intercept
Use Slopes to Identify Parallel Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.
Use Slopes to Identify Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

### 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?


# Graphs

## Find the Equation of a Line

How do online retailers know that ‘you may also like’ a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.
You are at an exciting point in your mathematical journey as the mathematics you are studying has interesting applications in the real world.
The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. Data is collected and graphed. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable.
To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.

### Find an Equation of the Line Given the Slope and y-Intercept

We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form,  Now, we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.
Sometimes, the slope and intercept need to be determined from the graph.

### Find an Equation of the Line Given the Slope and a Point

Finding an equation of a line using the slope–intercept form of the equation works well when you are given the slope and y-intercept or when you read them off a graph. But what happens when you have another point instead of the y-intercept?
We are going to use the slope formula to derive another form of an equation of the line. Suppose we have a line that has slope  and that contains some specific point  and some other point, which we will just call . We can write the slope of this line and then change it to a different form.
This format is called the point–slope form of an equation of a line.
We can use the point–slope form of an equation to find an equation of a line when we are given the slope and one point. Then we will rewrite the equation in slope–intercept form. Most applications of linear equations use the the slope–intercept form.

### Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.
We have two options so far for finding an equation of a line: slope–intercept or point–slope. Since we will know two points, it will make more sense to use the point–slope form.
But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.
We have seen that we can use either the slope–intercept form or the point–slope form to find an equation of a line. Which form we use will depend on the information we are given. This is summarized in .

### Find an Equation of a Line Parallel to a Given Line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point–slope equation.
First let’s look at this graphically.
The graph shows the graph of . We want to graph a line parallel to this line and passing through the point .
We know that parallel lines have the same slope. So the second line will have the same slope as. That slope is. We’ll use the notation  to represent the slope of a line parallel to a line with slope . (Notice that the subscript  looks like two parallel lines.)
The second line will pass through  and have . To graph the line, we start at and count out the rise and run. With  (or ), we count out the rise 2 and the run 1. We draw the line.
Do the lines appear parallel? Does the second line pass through ?
Now, let’s see how to do this algebraically.
We can use either the slope–intercept form or the point–slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point–slope form.

### Find an Equation of a Line Perpendicular to a Given Line

Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point–slope equation, like we did with parallel lines.
The graph shows the graph of . Now, we want to graph a line perpendicular to this line and passing through .
We know that perpendicular lines have slopes that are negative reciprocals. We’ll use the notation  to represent the slope of a line perpendicular to a line with slope . (Notice that the subscript ⊥ looks like the right angles made by two perpendicular lines.)
We now know the perpendicular line will pass through  with .
To graph the line, we will start at  and count out the rise  and the run 2. Then we draw the line.
Do the lines appear perpendicular? Does the second line pass through ?
Now, let’s see how to do this algebraically. We can use either the slope–intercept form or the point–slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point–slope form.
In , we used the point–slope form to find the equation. We could have looked at this in a different way.
We want to find a line that is perpendicular to  that contains the point . The graph shows us the line and the point .
We know every line perpendicular to a vetical line is horizontal, so we will sketch the horizontal line through .
Do the lines appear perpendicular?
If we look at a few points on this horizontal line, we notice they all have y-coordinates of . So, the equation of the line perpendicular to the vertical line  is .

### Key Concepts

1. To Find an Equation of a Line Given the Slope and a Point

2. To Find an Equation of a Line Given Two Points

3. To Write and Equation of a Line

4. To Find an Equation of a Line Parallel to a Given Line

5. To Find an Equation of a Line Perpendicular to a Given Line

### Practice Makes Perfect

Find an Equation of the Line Given the Slope and
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

### Mixed Practice

In the following exercises, find the equation of each line. Write the equation in slope–intercept form.

### 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?


# Graphs

## Graphs of Linear Inequalities

### Verify Solutions to an Inequality in Two Variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.
Do you remember that an inequality with one variable had many solutions? The solution to the inequality  is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See .
Similarly, inequalities in two variables have many solutions. Any ordered pair  that makes the inequality true when we substitute in the values is a solution of the inequality.

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

Now, we will look at how the solutions of an inequality relate to its graph.
Let’s think about the number line in  again. The point  separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See .
The solution to  is the shaded part of the number line to the right of .
Similarly, the line  separates the plane into two regions. On one side of the line are points with . On the other side of the line are the points with . We call the line  a boundary line.
For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not  is included in the solution:
Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in
Now, let’s take a look at what we found in . We’ll start by graphing the line , and then we’ll plot the five points we tested. See .
In  we found that some of the points were solutions to the inequality  and some were not.
Which of the points we plotted are solutions to the inequality ? The points  and  are solutions to the inequality . Notice that they are both on the same side of the boundary line .
The two points  and  are on the other side of the boundary line , and they are not solutions to the inequality . For those two points, .
What about the point ? Because , the point is a solution to the equation . So the point  is on the boundary line.
Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality . The point  clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality?
Any point you choose on the left side of the boundary line is a solution to the inequality . All points on the left are solutions.
Similarly, all points on the right side of the boundary line, the side with  and , are not solutions to . See .
The graph of the inequality  is shown in  below. The line  divides the plane into two regions. The shaded side shows the solutions to the inequality .
The points on the boundary line, those where , are not solutions to the inequality , so the line itself is not part of the solution. We show that by making the line dashed, not solid.

### Graph Linear Inequalities

Now, we’re ready to put all this together to graph linear inequalities.
The steps we take to graph a linear inequality are summarized here.
What if the boundary line goes through the origin? Then we won’t be able to use  as a test point. No problem—we’ll just choose some other point that is not on the boundary line.
Some linear inequalities have only one variable. They may have an x but no y, or a y but no x. In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

### Key Concepts

1. To Graph a Linear Inequality

### Section Exercises

### Practice Makes Perfect

Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities
In the following exercises, graph each linear inequality.

### 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?

### Chapter 4 Review Exercises

### Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
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, which ordered pairs are solutions to the given equations?
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 the following exercises, for each ordered pair, decide:
1. ⓐ Is the ordered pair a solution to the equation?
2. ⓑ Is the point on the line?
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 each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.

### Graphing with Intercepts

Identify the
In the following exercises, find the x- and y-intercepts.
Find the
In the following exercises, find the intercepts of each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.

### 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.
Use
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 each line with the given point and slope.
Solve Slope Applications
In the following exercises, solve these slope applications.

### Intercept Form of an Equation of a Line

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line
In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation .
Identify the Slope and y-Intercept from an Equation of a Line
In the following exercises, identify the slope and y-intercept of each line.
Graph a Line Using Its Slope and Intercept
In the following exercises, graph the line of each equation using its slope and y-intercept.
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope–Intercept
Use Slopes to Identify Parallel Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.
Use Slopes to Identify Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

### Find the Equation of a Line

Find an Equation of the Line Given the Slope and
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

### Graph Linear Inequalities

Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities
In the following exercises, graph each linear inequality.

### Practice Test

Find the slope of each line shown.
Graph the line for each of the following equations.
Find the equation of each line. Write the equation in slope–intercept form.
Graph each linear inequality.


# Systems of Linear Equations

## Introduction

An architect designing a home may have restrictions on both the area and perimeter of the windows because of energy and structural concerns. The length and width chosen for each window would have to satisfy two equations: one for the area and the other for the perimeter. Similarly, a banker may have a fixed amount of money to put into two investment funds. A restaurant owner may want to increase profits, but in order to do that he will need to hire more staff. A job applicant may compare salary and costs of commuting for two job offers.
In this chapter, we will look at methods to solve situations like these using equations with two variables.


# Systems of Linear Equations

## Solve Systems of Equations by Graphing

### Determine Whether an Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.
Now we will work with systems of linear equations, two or more linear equations grouped together.
We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.
An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.
A linear equation in two variables, like 2x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.
To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (x, y) that make both equations true. These are called the solutions to a system of equations.
To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.
Let’s consider the system below:
Is the ordered pair  a solution?
The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.
Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?
The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

### Solve a System of Linear Equations by Graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.
The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.
Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.
Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in :
For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.
The steps to use to solve a system of linear equations by graphing are shown below.
Both equations in  were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.
Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We’ll do this in .
Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.
In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.
If you write the second equation in  in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept.
When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept.

### Determine the Number of Solutions of a Linear System

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.
We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in  through  all had two intersecting lines. Each system had one solution.
A system with parallel lines, like , has no solution. What happened in ? The equations have coincident lines, and so the system had infinitely many solutions.
We’ll organize these results in  below:
Parallel lines have the same slope but different y-intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in .
The two lines have the same slope but different y-intercepts. They are parallel lines.
 shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.
Let’s take one more look at our equations in  that gave us parallel lines.
When both lines were in slope-intercept form we had:
Do you recognize that it is impossible to have a single ordered pair  that is a solution to both of those equations?
We call a system of equations like this an inconsistent system. It has no solution.
A system of equations that has at least one solution is called a consistent system.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent equations, they each have their own set of solutions. Intersecting lines and parallel lines are independent.
If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.
Let’s sum this up by looking at the graphs of the three types of systems. See  and .

### Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.

Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

### Key Concepts

1. To solve a system of linear equations by graphing

2. Determine the number of solutions from the graph of a linear system

3. Determine the number of solutions of a linear system by looking at the slopes and intercepts

4. Determine the number of solutions and how to classify a system of equations

5. Problem Solving Strategy for Systems of Linear Equations

### Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.
Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.
Solve Applications of Systems of Equations by Graphing 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.
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.


# Systems of Linear Equations

## Solving Systems of Equations by Substitution

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.
In this section, we will solve systems of linear equations by the substitution method.

### Solve a System of Equations by Substitution

We will use the same system we used first for graphing.
We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.
Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!
After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.
We’ll fill in all these steps now in .
If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in .
If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y.
In  it was easiest to solve for y in the first equation because it had a coefficient of 1. In  it will be easier to solve for x.
When both equations are already solved for the same variable, it is easy to substitute!
Be very careful with the signs in the next example.
In , it will take a little more work to solve one equation for x or y.
Look back at the equations in . Is there any way to recognize that they are the same line?
Let’s see what happens in the next example.

### Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.
Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?
In the , we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.
For  we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

### Key Concepts

1. Solve a system of equations by substitution

### Practice Makes Perfect

Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve Applications of Systems of Equations by Substitution
In the following exercises, translate to a system of equations and 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 reviewing this checklist, what will you do to become confident for all objectives?


# Systems of Linear Equations

## Solve Systems of Equations by Elimination

We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.
The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we’ll do with the elimination method, too, but we’ll have a different way to get there.

### Solve a System of Equations by Elimination

The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.
For any expressions a, b, c, and d,
To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.
Notice how that works when we add these two equations together:
The y’s add to zero and we have one equation with one variable.
Let’s try another one:
This time we don’t see a variable that can be immediately eliminated if we add the equations.
But if we multiply the first equation by −2, we will make the coefficients of x opposites. We must multiply every term on both sides of the equation by −2.
Now we see that the coefficients of the x terms are opposites, so x will be eliminated when we add these two equations.
Add the equations yourself—the result should be −3y = −6. And that looks easy to solve, doesn’t it? Here is what it would look like.
We’ll do one more:
It doesn’t appear that we can get the coefficients of one variable to be opposites by multiplying one of the equations by a constant, unless we use fractions. So instead, we’ll have to multiply both equations by a constant.
We can make the coefficients of x be opposites if we multiply the first equation by 3 and the second by −4, so we get 12x and −12x.
This gives us these two new equations:
When we add these equations,
the x’s are eliminated and we just have −29y = 58.
Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.
Now we’ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.
The steps are listed below for easy reference.
First we’ll do an example where we can eliminate one variable right away.
In , we will be able to make the coefficients of one variable opposites by multiplying one equation by a constant.
Now we’ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.
When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by its LCD.
In the Solving Systems of Equations by Graphing we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.

### Solve Applications of Systems of Equations by Elimination

Some applications problems translate directly into equations in standard form, so we will use the elimination method to solve them. As before, we use our Problem Solving Strategy to help us stay focused and organized.

### Choose the Most Convenient Method to Solve a System of Linear Equations

When you will have to solve a system of linear equations in a later math class, you will usually not be told which method to use. You will need to make that decision yourself. So you’ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.

### Key Concepts

1. To Solve a System of Equations by Elimination

### Practice Makes Perfect

Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
Solve Applications of Systems of Equations by Elimination
In the following exercises, translate to a system of equations and solve.
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

### 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?


# Systems of Linear Equations

## Solve Applications with Systems of Equations

Previously in this chapter we solved several applications with systems of linear equations. In this section, we’ll look at some specific types of applications that relate two quantities. We’ll translate the words into linear equations, decide which is the most convenient method to use, and then solve them.
We will use our Problem Solving Strategy for Systems of Linear Equations.

### Translate to a System of Equations

Many of the problems we solved in earlier applications related two quantities. Here are two of the examples from the chapter on Math Models.
1. The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
2. A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?
In that chapter we translated each situation into one equation using only one variable. Sometimes it was a bit of a challenge figuring out how to name the two quantities, wasn’t it?
Let’s see how we can translate these two problems into a system of equations with two variables. We’ll focus on Steps 1 through 4 of our Problem Solving Strategy.
We’ll do another example where we stop after we write the system of equations.

### Solve Direct Translation Applications

We set up, but did not solve, the systems of equations in  and  Now we’ll translate a situation to a system of equations and then solve it.

### Solve Geometry Applications

When we learned about Math Models, we solved geometry applications using properties of triangles and rectangles. Now we’ll add to our list some properties of angles.
The measures of two complementary angles add to 90 degrees. The measures of two supplementary angles add to 180 degrees.
If two angles are complementary, we say that one angle is the complement of the other.
If two angles are supplementary, we say that one angle is the supplement of the other.

### Solve Uniform Motion Applications

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was D = rt where D is the distance travelled, r is the rate, and t is the time.
Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.
Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.
Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.
 and  show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water b and the speed of the river current c.
In  the boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is b + c.
In  the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is .
We’ll put some numbers to this situation in .
Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in . A wind current in the same direction as the plane is flying is called a tailwind. A wind current blowing against the direction of the plane is called a headwind.

### Practice Makes Perfect

Translate to a System of Equations
In the following exercises, translate to a system of equations and solve the system.
Solve Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications In the following exercises, translate to a system of equations and solve.

### 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?


# Systems of Linear Equations

## Solve Mixture Applications with Systems of Equations

### Solve Mixture Applications

When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:
Using one variable meant that we had to relate the number of nickels and the number of dimes. We had to decide if we were going to let n be the number of nickels and then write the number of dimes in terms of n, or if we would let d be the number of dimes and write the number of nickels in terms of d.
Now that we know how to solve systems of equations with two variables, we’ll just let n be the number of nickels and d be the number of dimes. We’ll write one equation based on the total value column, like we did before, and the other equation will come from the number column.
For the first example, we’ll do a ticket problem where the ticket prices are in whole dollars, so we won’t need to use decimals just yet.
In  we’ll solve a coin problem. Now that we know how to work with systems of two variables, naming the variables in the ‘number’ column will be easy.
Some mixture applications involve combining foods or drinks. Example situations might include combining raisins and nuts to make a trail mix or using two types of coffee beans to make a blend.
Another application of mixture problems relates to concentrated cleaning supplies, other chemicals, and mixed drinks. The concentration is given as a percent. For example, a 20% concentrated household cleanser means that 20% of the total amount is cleanser, and the rest is water. To make 35 ounces of a 20% concentration, you mix 7 ounces (20% of 35) of the cleanser with 28 ounces of water.
For these kinds of mixture problems, we’ll use percent instead of value for one of the columns in our table.

### Solve Interest Applications

The formula to model interest applications is I = Prt. Interest, I, is the product of the principal, P, the rate, r, and the time, t. In our work here, we will calculate the interest earned in one year, so t will be 1.
We modify the column titles in the mixture table to show the formula for interest, as you’ll see in .

### Key Concepts

1. Table for coin and mixture applications

2. Table for concentration applications

3. Table for interest applications

### Practice Makes Perfect

Solve Mixture Applications
In the following exercises, translate to a system of equations and solve.
Solve Interest Applications
In the following exercises, translate to a system of equations and solve.

### Everyday Math

In the following exercises, translate to a system of equations and solve.

### 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?


# Systems of Linear Equations

## Graphing Systems of Linear Inequalities

### Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.
A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.
To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs  that make both inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

### Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.
Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in .

### Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.

### Key Concepts

1. To Solve a System of Linear Inequalities by Graphing

### Section Exercises

### Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and 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 reviewing this checklist, what will you do to become confident for all objectives?

### Chapter 5 Review Exercises

### Solve Systems of Equations by Graphing

Determine Whether an Ordered Pair is a Solution of a System of Equations.
In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing
In the following exercises, solve the following systems of equations by graphing.
Determine the Number of Solutions of a Linear System
In the following exercises, without graphing determine the number of solutions and then classify the system of equations.
Solve Applications of Systems of Equations by Graphing

### Solve Systems of Equations by Substitution

Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve Applications of Systems of Equations by Substitution
In the following exercises, translate to a system of equations and solve.

### Solve Systems of Equations by Elimination

Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
Solve Applications of Systems of Equations by Elimination
In the following exercises, translate to a system of equations and solve.
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

### Solve Applications with Systems of Equations

Translate to a System of Equations
In the following exercises, translate to a system of equations. Do not solve the system.
Solve Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications
In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications
In the following exercises, translate to a system of equations and solve.

### Solve Mixture Applications with Systems of Equations

Solve Mixture Applications
In the following exercises, translate to a system of equations and solve.
Solve Interest Applications
In the following exercises, translate to a system of equations and solve.

### Graphing Systems of Linear Inequalities

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

### Practice Test

In the following exercises, solve the following systems by graphing.
In the following exercises, solve each system of equations. Use either substitution or elimination.
In the following exercises, translate to a system of equations and solve.


# Polynomials

## Introduction

We have seen that the graphs of linear equations are straight lines. Graphs of other types of equations, called polynomial equations, are curves, like the outline of this suspension bridge. Architects use polynomials to design the shape of a bridge like this and to draw the blueprints for it. Engineers use polynomials to calculate the stress on the bridge’s supports to ensure they are strong enough for the intended load. In this chapter, you will explore operations with and properties of polynomials.