# Algebra

## Algebraic Expressions

### Learning Objectives

After completing this section, you should be able to:
1. Convert between written and symbolic algebraic expressions and equations.
2. Simplify and evaluate algebraic expressions.
3. Add and subtract algebraic expressions.
4. Multiply and divide algebraic expressions.
Algebraic expressions are the building blocks of algebra. While a numerical expression (also known as an arithmetic expression) like  can represent only a single number, an algebraic expression such as  can represent many different numbers. This section will introduce you to algebraic expressions, how to create them, simplify them, and perform arithmetic operations on them.

### Algebraic Expressions and Equations

Xavier and Yasenia have the same birthday, but they were born in different years. This year Xavier is 20 years old and Yasenia is 23, so Yasenia is three years older than Xavier. When Xavier was 15, Yasenia was 18. When Xavier will be 33, Yasenia will be 36. No matter what Xavier’s age is, Yasenia’s age will always be 3 years more.
In the language of algebra, we say that Xavier's age and Yasenia's age are variable and the 3 is a constant. The ages change, or vary, so age is a variable. The 3 years between them always stays the same or has the same value, so the age difference is the constant. In algebra, letters of the alphabet are used to represent variables. The letters most often used for variables are , , , , , and . Suppose we call Xavier's age . Then we could use  to represent Yasenia's age, as shown in the table below.
To write algebraically, we need some symbols as well as numbers and variables. The symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division are summarized in , along with words we use for the operations and the result.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
1. The sum  5  3 means add 5 plus 3, which we write as .
2. The difference  9  2 means subtract 9 minus 2, which we write as .
3. The product  4  8 means multiply 4 times 8, which we can write as .
4. The quotient  20  5 means divide 20 by 5, which we can write as .
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have expressions and equations.  and  used expressions. An expression is like an English phrase. Notice that the English phrases do not form a complete sentence because the phrase does not have a verb. The following table has examples of expressions, which are numbers, variables, or combinations of numbers and variables using operation symbols.
An equation is two expressions linked with an equal sign (the symbol =). When two quantities have the same value, we say they are equal and connect them with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. So,  is read “ is equal to .” The following table has some examples of equations.

### Simplifying and Evaluating Algebraic Expressions

To simplify an expression means to do all the math possible. For example, to simplify  we would first multiply  to get 8 and then add 1 to get 9. We have introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. Consider . Do you add first or multiply first? Do you get different answers?
Early on, mathematicians realized the need to establish some guidelines when performing arithmetic operations to ensure that everyone would get the same answer. Those guidelines are called the order of operations and are listed in the table below.
In the last example, we simplified expressions using the order of operations. Now we'll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

### Operations of Algebraic Expressions

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Examples of terms are 7, , 5, 9, and . The constant that multiplies the variable is called the coefficient. Think of the coefficient as the number in front of the variable. Consider the algebraic expressions 5, which has a coefficient of 5, and 9, which has a coefficient of 9. If there is no number listed in front of the variable, then the coefficient is 1 since .
Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms. If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.
Before looking at multiplying algebraic expressions we look at the Distributive Property, which says that to multiply a sum, first you multiply each term in the sum and then you add the products. For example,  can also be solved as . If we use a variable, then .
We can extended this example to , which can also be solved as . If we use variables, then .
In 2012, Andrew Hacker wrote an opinion piece in the New York Times Magazine suggesting that teaching algebra in high school was a waste of time. Keith Devlin, a British mathematician, was asked to comment on Hacker's article by his students in his Stanford University Continuing Studies course "Mathematics: Making the Invisible Visible" on iTunes University. Devlin concludes that Hacker was displaying his ignorance of what algebra is.

### Check Your Understanding

### Key Terms

1. variable
2. constant
3. expression
4. equation
5. equal sign
6. term
7. coefficient
8. like terms
9. Distributive Property

### Key Concepts

1. Algebra is useful because it allows us to understand many situations in real life by modeling them with expressions.
2. Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
3. Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
4. Algebraic expressions are often simplified and evaluated using the four arithmetic operations.

### Videos

1. Q&A: Why We Teach Algebra

### Formulas

1.  Distributive Property:


# Algebra

## Linear Equations in One Variable with Applications

### Learning Objectives

After completing this section, you should be able to:
1. Solve linear equations in one variable using properties of equations.
2. Construct a linear equation to solve applications.
3. Determine equations with no solution or infinitely many solutions.
4. Solve a formula for a given variable.
In this section, we will study linear equations in one variable. There are several real-world scenarios that can be represented by linear equations: taxi rentals with a flat fee and a rate per mile; cell phone bills that charge a monthly fee plus a separate rate per text; gym memberships with a monthly fee plus a rate per class taken; etc. For example, if you join your local gym at $10 per month and pay $5 per class, how many classes can you take if your gym budget is $75 per month?

### Linear Equations and Applications

Solving any equation is like discovering the answer to a puzzle. The purpose of solving an equation is to find the value or values of the variable that makes the equation a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle! There are many types of equations that we will learn to solve. In this section, we will focus on a linear equation, which is an equation in one variable that can be written as
where  and  are real numbers and , such that  is the coefficient of  and  is the constant.
To solve a linear equation, it is a good idea to have an overall strategy that can be used to solve any linear equation. In the Example 5.12, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.
In , we used both the addition and division property of equations. All the properties of equations are summarized in table below. Basically, what you do to one side of the equation, you must do to the other side of the equation to preserve equality.
In Algebraic Expressions, you translated an English sentence into an equation. In this section, we take that one step further and translate an English paragraph into an equation, and then we solve the equation. We can go back to the opening question in this section: If you join your local gym at $10 per month and pay $5 per class, how many classes can you take if your gym budget is $75 per month? We can create an equation for this scenario and then solve the equation (see ).

### Linear Equations with No Solutions or Infinitely Many Solutions

Every linear equation we have solved thus far has given us one numerical solution. Now we'll look at linear equations for which there are no solutions or infinitely many solutions.

### Solving a Formula for a Given Variable

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be able to manipulate formulas and solve for specific variables.
To solve a formula for a specific variable means to isolate that variable on one side of the equal sign with a coefficient of 1. All other variables and constants are on the other side of the equal sign. To see how to solve a formula for a specific variable, we will start with the distance, rate, and time formula.

### Check Your Understanding

### Key Terms

1. linear equation

### Key Concepts

1. Solving linear equations means discovering what the value of the variable in a linear equation represents in the given conditions.
2. When solving a linear equation, most often you will have one solution; however, a linear equation may have no solutions or infinitely many solutions.

### Videos

1. Solving for a Variable in an Equation


# Algebra

## Linear Inequalities in One Variable with Applications

### Learning Objectives

After completing this section, you should be able to:
1. Graph inequalities in one variable.
2. Solve linear inequalities in one variable.
3. Construct a linear inequality to solve applications.
In this section, we will study linear inequalities in one variable. Inequalities can be used when the possible values (answers) in a certain situation are numerous, not just a few, or when the exact value (answer) is not known but it is known to be within a range of possible values. There are many real-world scenarios that can be represented by linear inequalities. For example, consider the survey of the mayoral election in  Surveys and polls are usually conducted with only a small group of people. The margin of error indicates a range of how the actual group of voters would vote given the results of the survey. This range can be expressed using inequalities.
Another example involves college tuition. Say a local community college charges $113 per credit hour. You budget $1,500 for tuition this fall semester. What are the number of credit hours that you could take this fall? Since this answer could be many different values, it can be expressed as an inequality.

### Graphing Inequalities on the Number Line

In Algebraic Expressions, we introduced equality and the  symbol. In this section, we look at inequality and the symbols , , , and . The table below summarizes the symbols and their meaning.

Suppose you had the inequality statement . What possible number or numbers would make the inequality  true? If you are thinking, " could be 4," that's correct, but  could also be 5, 6, 37, 1 million, or even 3.001. The number of solutions is infinite; any number greater than 3 is a solution to the inequality .
Rather than trying to list all possible solutions, we show all the solutions to the inequality  on the number line. All the numbers to the right of 3 on the number line are shaded, to show that all numbers greater than 3 are solutions. At the number 3 itself, an open parenthesis is drawn, since the number 3 is not part of the solutions of .
We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express  as . The symbol  is read as "infinity." Infinity is not an actual number.  shows both the number line and the interval notation for .
We used the left parenthesis symbol to show that the endpoint of the inequality is not included. Parentheses are used when the endpoints are not included as a possible answer to the inequality. The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.
The inequality  means all numbers less than or equal to 1. To illustrate that solution on a number line, we first put a bracket at ; brackets are used when the endpoint is included. We then shade in all the numbers to the left of 1, to show that all numbers less than one are solutions. There is no lower end to those numbers. We write  in interval notation as . The symbol  is read as "negative infinity."  shows both the number line and interval notation for .
 summarizes the general representations in both number line form and interval notation of solutions for , , , and .

### Solving Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms  or  where , , and  are all real numbers.
When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities. We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example, we know that 2 is less than 4, i.e., . If we add 6 to both sides of this inequality, we still have a true statement:
The same would happen if we subtracted 6 from both sides of the inequality; the statement would stay true:
Notice that the inequality signs stayed the same. This leads us to the Addition and Subtraction Properties of Inequality.
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality the same. But what happens to an inequality when we divide or multiply both sides by a number? Let's first multiply and divide both sides by a positive number, starting with an inequality we know is true, . We will multiply and divide this inequality by 5:
The inequality signs stayed the same. Does the inequality stay the same when we divide or multiply by a negative number? Let's use our inequality  to find out, multiplying it and dividing it by :
Notice that when we filled in the inequality signs, the inequality signs reversed their direction in order to make it true! To summarize, when we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses. This gives us the Multiplication and Division Property of Inequality.
To summarize, when we divide or multiply an inequality by:

### Solving Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. 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.

### Check Your Understanding

### Key Terms

1. linear inequality
2. Addition and Subtraction Property of Linear Inequalities
3. Multiplication and Division Property of Linear Inequalities

### Key Concepts

1. Inequalities can be used when the possible values (answers) in a certain situation are numerous, or when the exact value (answer) is not known, but it is known to be within a range of possible values.
2. Linear inequalities can be represented using a number line or using interval notation.

### Formulas

1. For any numbers , , and  if , then  and .
2. For any numbers , , and , if , then  and .
3. For any numbers , , and , multiply or divide by a positive:
     if  and , then  and
     if  and , then  and
     multiply or divide by a negative:
     if  and , then  and
     if  and , then  and


# Algebra

## Ratios and Proportions

### Learning Objectives

After completing this section, you should be able to:
1. Construct ratios to express comparison of two quantities.
2. Use and apply proportional relationships to solve problems.
3. Determine and apply a constant of proportionality.
4. Use proportions to solve scaling problems.
Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from , we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.

### Constructing Ratios to Express Comparison of Two Quantities

Note there are three different ways to write a ratio, which is a comparison of two numbers that can be written as:  to  OR  OR the fraction . Which method you use often depends upon the situation. For the most part, we will want to write our ratios using the fraction notation. Note that, while all ratios are fractions, not all fractions are ratios. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.

### Using and Applying Proportional Relationships to Solve Problems

Using proportions to solve problems is a very useful method. It is usually used when you know three parts of the proportion, and one part is unknown. Proportions are often solved by setting up like ratios. If  and  are two ratios such that  then the fractions are said to be proportional. Also, two fractions  and  are proportional  if and only if .

### Determining and Applying a Constant of Proportionality

In the last example, we were given that  cups of flour could make 60 cookies; we then calculated that  cups of flour would make 1,020 cookies, and 720 cookies could be made from 27 cups of flour. Each of those three ratios is written as a fraction below (with the fractions converted to decimals). What happens if you divide the numerator by the denominator in each?
The quotients in each are exactly the same! This number, determined from the ratio of cups of flour to cookies, is called the constant of proportionality. If the values  and  are related by the equality  then  is the constant of proportionality between  and . Note since  then  and
One piece of information that we can derive from the constant of proportionality is a unit rate. In our example (cups of flour divided by cookies), the constant of proportionality is telling us that it takes 0.0375 cups of flour to make one cookie. What if we had performed the calculation the other way (cookies divided by cups of flour)?
In this case, the constant of proportionality  is telling us that  cookies can be made with one cup of flour. Notice in both cases, the "one" unit is associated with the denominator. The constant of proportionality is also useful in calculations if you only know one part of the ratio and wish to find the other.

### Using Proportions to Solve Scaling Problems

Ratio and proportions are used to solve problems involving scale. One common place you see a scale is on a map (as represented in ). In this image, 1 inch is equal to 200 miles. This is the scale. This means that 1 inch on the map corresponds to 200 miles on the surface of Earth. Another place where scales are used is with models: model cars, trucks, airplanes, trains, and so on. A common ratio given for model cars is 1:24—that means that 1 inch in length on the model car is equal to 24 inches (2 feet) on an actual automobile. Although these are two common places that scale is used, it is used in a variety of other ways as well.

### Check Your Understanding

### Key Terms

1. ratio
2. proportion
3. constant of proportionality
4. scale
5. construct ratios
6. solve proportions
7. use proportions to solve scaling problems

### Key Concepts

1. A ratio is a comparison of two numbers. The ratio of two numbers  and  can be written as:  to  OR : OR the fraction /.
2. All fractions are ratios, but not all ratios are fractions. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.
3. When two ratios are equal, we say they are in proportion or are proportional.
4. Setting up proportions allows us to solve many various situations where three of the four values of the proportion are known.


# Algebra

## Graphing Linear Equations and Inequalities

### Learning Objectives

After completing this section, you should be able to:
1. Graph linear equations and inequalities in two variables.
2. Solve applications of linear equations and inequalities.
In this section, we will learn how to graph linear equations and inequalities. There are several real-world scenarios that can be represented by graphs of linear inequalities. Think of filling your car up with gasoline. If gasoline is $3.99 per gallon and you put 10 gallons in your car, you will pay $39.90. Your friend buys 15 gallons of gasoline and pays $59.85. You can plot these points on a coordinate system and connect the points with a line to create the graph of a line. You'll learn to do both in this section.

### Plotting 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 -plane or the “coordinate plane.”
The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the -axis. The vertical number line is called the -axis. 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 -coordinate of the point, and the second number is the -coordinate of the point. The phrase "ordered pair" means that the order is important. At the point where the axes cross and where both coordinates are zero, the ordered pair is . The point  has a special name. It is called the origin.
We use the coordinates to locate a point on the -plane. Let's plot the point  as an example. First, locate 1 on the -axis and lightly sketch a vertical line through . Then, locate 3 on the -axis and sketch a horizontal line through . Now, find the point where these two lines meet—that is the point with coordinates . See .

Notice that the vertical line through  and the horizontal line through  are not part of the graph. The dotted lines are just used to help us locate the point . When one of the coordinates is zero, the point lies on one of the axes. In , the point  is on the -axis and the point (−2, 0) is on the -axis.

### Graphing Linear Equations 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. But equations can have more than one variable. Equations with two variables may be of the form . An equation of this form, where  and  are both not zero, is called a linear equation in two variables. 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.
Step 1: Add  to both sides.
Step 2: Simplify.
Step 3: Put 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 of a linear equation. 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.
We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the linear equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.
 A graph is a visual representation of all the solutions of a linear equation. The line shows you all the solutions to that linear equation. Every point on the line is a solution of that linear equation. And every solution of the linear equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions! The graph of a linear equation  is a straight line.

The steps to take when graphing a linear equation by plotting points are:
Step 1: Find three points whose coordinates are solutions to the equation. Organize them in a table.
Step 2: Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
Step 3: Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
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.

### Solving Applications Using Linear Equations in Two Variables

Many fields use linear equalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

### Graphing Linear Inequalities

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables that can be written in one of the following forms: , , , and  where  and  are not both zero. We will look at linear inequalities in two variables, which are very similar to linear equations in two variables.
Like linear equations, linear inequalities in two variables have many solutions. Any ordered pair (, ) that makes an inequality true when we substitute in the values is a solution to a linear inequality.
Let us think about . 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 .
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 show whether or not it the line is included in the solution.
Now, let us take a look at what we found in Example 5.43. We will start by graphing the line , and then we will plot the five points we tested, as graphed 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 , but not a solution to the inequality . So, the point  is on the boundary line.
Let us take another point above the boundary line and test whether or not it is a solution to the inequality . The point  clearly looks to be above the boundary line, doesn’t it? Is it a solution to the inequality?

Yes,  is a solution to . Any point you choose above the boundary line is a solution to the inequality . All points above the boundary line are solutions. Similarly, all points below the boundary line, the side with  and , are not solutions to , as shown in .
The graph of the inequality  is shown in . 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 boundary line dashed, not solid.

### Solving Applications Using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

### Check Your Understanding

### Key Terms

1. ordered pair
2. origin
3. points on the axes
4. linear equation in two variables
5. standards form of a linear equation
6. solution
7. linear inequality in two variables
8. solution to a linear inequality
9. boundary line

### Key Concepts

1. Linear equations can be represented graphically on a rectangular coordinate system.
2. Solving linear equations in two variables means finding the point where two lines intersect. There are three possibilities: The lines intersect at exactly one point; the lines do not intersect (they are parallel); or the lines intersect everywhere (they are the same line).
3. Solving linear inequalities in two variables means finding a region of possible answers. Every point in this region will make both inequalities true statements.
4. Plotting points is a standard way to help graph linear equations and linear inequalities.

### Videos

1. Graphing Linear Inequalities in Two Variables


# Algebra

## Quadratic Equations with Two Variables with Applications

### Learning Objectives

After completing this section, you should be able to:
1. Multiply binomials.
2. Factor trinomials.
3. Solve quadratic equations by graphing.
4. Solve quadratic equations by factoring.
5. Solve quadratic equations using square root method.
6. Solve quadratic equations using the quadratic formula.
7. Solve real world applications modeled by quadratic equations.
In this section, we will discuss quadratic equations. There are several real-world scenarios that can be represented by the graph of a quadratic equation. Think of the Gateway Arch in St. Louis, Missouri. Both ends of the arch are 630 feet apart and the arch is 630 feet tall. You can plot these points on a coordinate system and create a parabola to graph the quadratic equation.

### Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant, or the product of a constant and one or more variables. When it is of the form , where  is a constant and  is a positive whole number, it is called a monomial. Some examples of monomial are 8, , , and .
A monomial or two or more monomials combined by addition or subtraction is a polynomial. Some examples include: , , and . Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term (examples: 14, , , and ). A binomial has exactly two terms (examples: , , , and ), and a trinomial has exactly three terms (examples: , , , and ).
Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

### Multiply Binomials

Recall multiplying algebraic expressions from Algebraic Expressions. In this section, we will continue that work and multiply binomials as well. We can use an area model to do multiplication.

### Factoring Trinomials

We’ve just covered how to multiply binomials. Now you will need to “undo” this multiplication—to start with the product and end up with the factors. Let us review an example of multiplying binomials to refresh your memory.
To factor the trinomial means to start with the product, , and end with the factors, . You need to think about where each of the terms in the trinomial came from. The first term came from multiplying the first term in each binomial. So, to get  in the product, each binomial must start with an .
The last term in the trinomial came from multiplying the last term in each binomial. So, the last terms must multiply to 6. What two numbers multiply to 6? The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use? Consider the middle term. It came from adding the outer and inner terms. So the numbers that must have a product of 6 will need a sum of 5.
We’ll test both possibilities and summarize the results in the following table, which will be very helpful when you work with numbers that can be factored in many different ways.
We see that 2 and 3 are the numbers that multiply to 6 and add to 5. We have the factors of . They are .
You can check if the factors are correct by multiplying. Looking back, we started with , which is of the form , where  and . We factored it into two binomials of the form  and .
To get the correct factors, we found two number  and  whose product is  and sum is . With the area model (), start with an empty box and then put in the  term and .
Continue by putting in two terms that add up to  and  ():
Then you find the terms of the binomials on the top and side ():

### Solving Quadratic Equations by Graphing

We have already solved and graphed linear equations in Graphing Linear Equations and Inequalities, equations of the form . In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. The following are some examples of quadratic equations:
The last equation does not appear to have the variable squared, but when we simplify the expression on the left, we will get . The general form of a quadratic equation is , where  and  are real numbers, with . Remember that a solution of an equation is a value of a variable that makes a true statement when substituted into the equation. The solutions of quadratic equations are the values of the variables that make the quadratic equation  true.
To solve quadratic equations, we need methods different than the ones we used in solving linear equations. We will start by solving a quadratic equation from its graph. Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations. Let us look first at graphing the quadratic equation . We will choose integer values of  between  and 2 and find their  values, as shown in the table below.
Notice when we let  and , we got the same value for .
The same thing happened when we let  and . Now, we will plot the points to show the graph of . See .
The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this. When  the solution to the quadratic  is 0 because  at .

### Solving Quadratic Equations by Factoring

Another way of solving quadratic equations is by factoring. We will use the Zero Product Property that says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.

### Solving Quadratic Equations Using the Square Root Property

We just solved some quadratic equations by factoring. Let us use factoring to solve the quadratic equation .
The solution is read as “ is equal to positive or negative 3.”
What happens when we have an equation like ? Since 7 is not a perfect square, we cannot solve the equation by factoring. These equations are all of the form . We define the square root of a number in this way: If , then  is a square root of . This leads to the Square Root Property.

### Solving Quadratic Equations Using the Quadratic Formula

This last method we will look at for solving quadratic equations is the quadratic formula. This method works for all quadratic equations, even the quadratic equations we could not factor! To use the quadratic formula, we substitute the values of , , and  into the expression on the right side of the formula. Then, we do all the math to simplify the expression. The result gives the solution(s) to the quadratic equation.

### Solving Real-World Applications Modeled by Quadratic Equations

There are problem solving strategies that will work well for applications that translate to quadratic equations. Here’s a problem-solving strategy to solve word problems:
Step 1: Read the problem. Make sure all the words and ideas are understood.
Step 2: Identify what we are looking for.
Step 3: Name what we are looking for. Choose a variable to represent that quantity.
Step 4: Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
Step 5: Solve the equation using good algebra techniques.
Step 6: Check the answer in the problem and make sure it makes sense.
Step 7: Answer the question with a complete sentence.
Were you surprised by the pair of negative integers that is one of the solutions? In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.

### Check Your Understanding

### Key Terms

1. monomial
2. polynomial
3. binomial
4. trinomial
5. quadratic equation
6. Zero Product Property

### Key Concepts

1. A quadratic equation is an algebraic equation where the highest power (degree) of the equation is two.
2. To solve a quadratic equation is to find the value(s) that when substituted in for the variables, will make the equation equal to zero.
3. There can be two, one, or no solutions to any quadratic equation.
4. There are several methods to solve a quadratic equation. These methods include factoring quadratic equations, graphic quadratic equations, using the square root method, and using the quadratic formula.

### Videos

1.  Factoring with the Box Method (Area Model)
2. Solving Quadratics with the Zero Property
3. Solving Quadratics with the Quadratic Formula


# Algebra

## Functions

### Learning Objectives

After completing this section, you should be able to:
1. Use function notation.
2. Determine if a relation is a function with different representations.
3. Apply the vertical line test.
4. Determine the domain and range of a function.
In this section, we will learn about relations and functions. As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number, and our birthday are matched to our name. There is a relationship between our name and each of those items. When your teacher gets their class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.
(Student name, Student ID #)
The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation. In general terms, a relation is any set of ordered pairs, (). All the -values in the ordered pairs together make up the domain. All the -values in the ordered pairs together make up the range.
There are many situations similar to the student's name and student ID # where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation. A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each -value is matched with only one -value.
Let us look at the relation between your friends and their birthdays in . Every friend has a birthday, but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24 as their birthday and that June and Liz share August 2. Since each person has exactly one birthday, the relation is a function.

### Use Function Notation

It is very convenient to name a function; most often functions are named , , , , , or . In any function, for each -value from the domain, we get a corresponding -value in the range. In the function , we write this range value  as (). This notation () is called function notation and is read "f of " or "the value of f at ." In this case the parentheses do not indicate multiplication.
We call  the independent variable as it can be any value in the domain. We call  the dependent variable as its value depends on . Much like when you first encountered the variable , function notation may be rather unsettling. But the more you use the notation, the more familiar you become with the notation, and the more comfortable you will be with it.
Let’s review the equation . To find the value of  when , we know to substitute  into the equation and then simplify.

The value of the function at  is 3. We do the same thing using function notation, the equation  can be written as . To find the value when , we write:

The value of the function at  is 3. This process of finding the value of  for a given value of  is called evaluating the function.

### Determining If a Relation Is a Function with Different Representations

We can determine whether a relation is a function by identifying the input and the output values. If each input value leads to only one output value, classify the relation as a function. If any input value leads to two or more outputs, do not classify the relation as a function.
We will review three different representations of relations and determine if they are functions: ordered pairs, mapping, and equations.
A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range. Consider the example of the relation between your friends and their birthdays used in . In this particular example, the domain is the set of people’s names, and the range is the set of their birthdays. This mapping was a function because everybody’s name maps to exactly one birthday.
In algebra functions will usually be represented by an equation. It is easiest to see if the equation is a function when it is solved for . If each value of  results in only one value of , then the equation defines a function.

### Applying the Vertical Line Test

We reviewed how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping, or an equation. We will now cover how to tell if a graph is that of a function.
An ordered pair  is a solution of a linear equation, if the equation is a true statement when the -values and -values of the ordered pair are substituted into the equation. The graph of a linear equation is a straight line where every point on the line is a solution of the equation, and every solution of this equation is a point on this line.  we can see that in the graph of the equation , for every -value there is only one -value, as shown in the accompanying table.
A relation is a function if every element of the domain has exactly one value in the range. The relation defined by the equation  is a function. If we look at the graph, each vertical dashed line only intersects the solid line at one point. This makes sense as in a function, for every -value there is only one -value. If the vertical line hit the graph twice, the -value would be mapped to two -values, and so the graph would not represent a function. This leads us a graphical method of determining functions called the vertical line test, which states that a set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

### Determining the Domain and Range of a Function

For the function  is the independent variable as it can be any value in the domain, and  is the dependent variable since its value depends on . For the function , the values of  make up the domain and the values of  make up the range.

### Check Your Understanding

### Key Terms

1. relation
2. domain
3. function
4. mapping
5. vertical line test

### Key Concepts

1. A relation is any set of ordered pairs . All of the -values of the set are the domain, and all of the -values of the set are the range.
2. A relation is a function if each -value in the domain is assigned to exactly one element in the range. A -value in the range can have more than one -value assigned to it; but each -value can only be assigned to one -value.
3. For the function  is the name of the function,  is the domain value variable, and  is the range value variable.
4. The vertical line test is a test that can be done on the graph of a relation to determine if it is a function.

### Videos

1. Relations and Functions
2. Domain and Range on Graphs


# Algebra

## Graphing Functions

### Learning Objectives

After completing this module, you should be able to:
1. Graph functions using intercepts.
2. Compute slope.
3. Graph functions using slope and -intercept.
4. Graph horizontal and vertical lines.
5. Interpret graphs of functions.
6. Model applications using slope and -intercept.
In this section, we will expand our knowledge of graphing by graphing linear functions. There are many real-world scenarios that can be represented by graphs of linear functions. Imagine a chairlift going up at a ski resort. The journey a skier takes travelling up the chairlift could be represented as a linear function with a positive slope. The journey a skier takes down the slopes could be represented by a linear function with a negative slope.

### Graphing Functions Using Intercepts

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 -axis and the -axis. These points are called the intercepts of a line. Let us review the graphs of the lines in .
The table below lists where each of these lines crosses the - and -axis. Do you see a pattern? For each line, the -coordinate of the point where the line crosses the -axis is zero. The point where the line crosses the -axis has the form   and is called the -intercept of the line. The -intercept occurs when  is zero. In each line, the -coordinate of the point where the line crosses the -axis is zero. The point where the line crosses the -axis has the form  and is called the -intercept of the line. The -intercept occurs when  is zero.

### Computing Slope

When graphing 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. In mathematics, the measure of the steepness of a line is called the slope of the line. To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices of the triangle and one side is horizontal and one side is vertical. Next, we measure or calculate 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.
We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope () is a rate of change.
The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of plumbing pipes, and the steepness of stairs are all applications of slope. As you ski or jog down a hill, you definitely experience slope.
Sometimes we will need to find the slope of a line between two points when we don’t have a graph to measure the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing. First, 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 such points. For example, (, ) would be said aloud as “ sub 1,  sub 1” and (, ) read “ sub 2,  sub 2.” The “sub” is a short way of saying “subscript.” We will use (, ) to identify the first point and (, ) to identify the second point in our slope equation. If we had more than two points, (if we were finding more than one slope), we could use (, ), (, ), and so on.
Let’s review 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 , as shown in .
On the graph, we count the rise of 3 and the run of 5. Notice on the graph that that (, ) is the point  and (, ) is the point . The rise can be found by subtracting the -coordinates, 6 and 3, and the run can be found by subtracting the -coordinates 7 and 2.
We have shown that  is really another version of . We can use this formula to find the slope of a line.

### Graphing Functions Using Slope and -Intercept

We have graphed linear equations by plotting points and using intercepts. Once we see how an equation in slope-intercept form and its graph are related, we will have one more method we can use to graph lines. Review the graph of the equation  in  and find its slope and -intercept.
The vertical and horizontal lines in the graph show us the rise is 1 and the run is 2, respectively.
Substituting into the slope formula:
The -intercept is . Look at the equation of this line.
Look at the slope and -intercept.
When a linear equation is solved for , the coefficient of the  term is the slope and the constant term is the -coordinate of the -intercept. We say that the equation  is in slope-intercept form. Sometimes the slope-intercept form is called the -form.

### Graphing Horizontal and Vertical Lines

Some linear equations have only one variable. They may have just  without the , or just  without an . This changes how we make a table of values to get the points to plot. Let us 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 the value of  is, the value of  is always . 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 will use 1, 2, and 3 for the -coordinates in the table below.
Plot the points from the table and connect them with a straight line (). Notice that we have graphed a vertical line.
What is the slope? If we take the two points  and  then the rise is 2 and the run is 0.
Using the slope formula we get:
The slope is undefined since division by zero is undefined. We say that the slope of the vertical line  is undefined. The slope of any vertical line  (where  is any number) will be undefined.
What if the equation has  but no ? Let’s graph the equation . This time the -value is a constant, so in this equation,  does not depend on . Fill in 4 for all the  values in the table below and then choose any values for . We will use 0, 2, and 4 for the -coordinates.
In , we have graphed a horizontal line passing through the -axis at 4.
What is the slope? If we take the two points  and  then the rise is 0 and the run is 2. Using the slope formula, we get . The slope of the horizontal line  is 0. The slope of any horizontal line  (where  is any number) will be 0. When the -coordinates are the same, the rise is 0.
The table below summarizes all the methods we have used to graph lines.

### Interpreting Graphs of Functions

An important yet often overlooked area in algebra involves interpreting graphs. Oftentimes in math classes, students are given mathematical functions and can make graphs to represent them. But the interpretation of graphs is a more applicable skill to the real world. Being able to “read” a graph—understanding its domain and range, what the intercepts mean, and what the slope (or curve) means— that's a real-world skill.

### Modeling Applications Using Slope and -Intercept

Many real-world applications are modeled by linear equations. We will review a few applications here so you can understand how equations written in slope-intercept form relate to real-world situations. Usually when a linear equation model uses real-world data, different letters are used for the variables instead of using only  and . The variable names often remind us of what quantities are being measured. Also, we often need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.

### Check Your Understanding

### Key Terms

1. intercepts of a line
2. slope
3. slope-intercept form

### Key Concepts

1. Every linear function can be graphically represented by a unique line that shows all the solutions of the equation.
2. The points where the graph of a line intersects the -axis and -axis are called the intercepts of the line.
3. Most lines will have one -intercept and one -intercept. Only if the line is straight vertical (no -intercept) or straight horizontal (no -intercept) will it not have both intercepts. Note that a line that is straight vertical is not a function, but a line that is straight horizontal is a function.
4. Since any two points determine a straight line, any linear function can be graphed if both intercepts are known.
5. The slope of a linear function is the ratio of the vertical change divided by the horizontal change. It is often referred to as .
6. A formula for finding the slope of linear functions is  for any two points of the linear function  and .

### Formulas

1.  To calculate slope , use the formula
   , where the rise measures the vertical change and the run measures the horizontal change.
2.  To find the slope of the line between two points  and , use the formula


# Algebra

## Systems of Linear Equations in Two Variables

### Learning Objectives

After completing this section, you should be able to:
1. Determine and show whether an ordered pair is a solution to a system of equations.
2. Solve systems of linear equations using graphical methods.
3. Solve systems of linear equations using substitution.
4. Solve systems of linear equations using elimination.
5. Identify systems with no solution or infinitely many solutions.
6. Solve applications of systems of linear equations.
In this section, we will learn how to solve systems of linear equations in two variables. There are several real-world scenarios that can be represented by systems of linear equalities. Suppose two friends, Andrea and Bart, go shopping at a farmers market to buy some vegetables. Andrea buys 2 tomatoes and 4 cucumbers and spends $2.00. Bart buys 4 tomatoes and 5 cucumbers and spends $2.95. What is the price of each vegetable?

### Determining If an Ordered Pair Is a Solution to a System of Equations

When we solved linear equations in Linear Equations in One Variable with Applications and Linear Inequalities in One Variable with Applications, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear equations.
In this section, we will focus our work on systems of two linear equations in two unknowns (variables) and applications of systems of linear 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, such as , 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 (, ) that make both equations true. These are called the solutions of 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.

### Solving Systems of Linear Equations Using Graphical Methods

We will use three methods to solve a system of linear equations. The first method we will 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 points the lines have in common, we will find the solution to the system.
Most linear equations in one variable have one solution; but for some equations called contradictions, there are 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 .
Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end you will 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 here.
Step 1: Graph the first equation.
Step 2: Graph the second equation on the same rectangular coordinate system.
Step 3: Determine whether the lines intersect, are parallel, or are the same line.
Step 4: Identify the solution to the system.
If the lines intersect, identify the point of intersection. This is the solution to the system.
If the lines are parallel, the system has no solution.
If the lines are the same, the system has an infinite number of solutions.
Step 5: Check the solution in both equations.

### Solving Systems of Linear Equations Using Substitution

We will now solve systems of linear equations by the substitution method. We will use the same system we used for graphing.
We will first solve one of the equations for either  or . 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. This process is summarized here:
Step 1: Solve one of the equations for either variable.
Step 2: Substitute the expression from Step 1 into the other equation.
Step 3: Solve the resulting equation.
Step 4: Substitute the solution in Step 3 into either of the original equations to find the other variable.
Step 5: Write the solution as an ordered pair.
Step 6: Check that the ordered pair is a solution to both original equations.

### Solving Systems of Linear Equations Using 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.
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 , , , and :
if
and
then .
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 ’s add to zero and we have one equation with one variable. Let us try another one:
This time we do not see a variable that can be immediately eliminated if we add the equations. But if we multiply the first equation by , we will make the coefficients of  opposites. We must multiply every term on both sides of the equation by .
Then rewrite the system of equations.
Now we see that the coefficients of the  terms are opposites, so  will be eliminated when we add these two equations.
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. Here’s a summary of using the elimination method:
Step 1: Write both equations in standard form. If any coefficients are fractions, clear them.
Step 2: Make the coefficients of one variable opposites.
Decide which variable you will eliminate.
Multiply one or both equations so that the coefficients of that variable are opposites.
Step 3: Add the equations resulting from Step 2 to eliminate one variable.
Step 4: Solve for the remaining variable.
Step 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.
Step 6: Write the solution as an ordered pair.
Step 7: Check that the ordered pair is a solution to both original equations.

### Identifying Systems with No Solution or Infinitely Many Solutions

In all the systems of linear equations so far, the lines intersected, and the solution was one point. In  and , we will look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.
In the previous example, if you write the second equation in slope-intercept form, you may recognize that the equations have the same slope and same -intercept. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system. We say the two lines are coincident. Coincident lines have the same slope and same -intercept. A system of equations that has at least one solution is called a consistent system. A system with parallel lines has no solution. We call a system of equations like this an inconsistent system. It has no solution.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent, 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 us sum this up by looking at the graphs of the three types of systems. See  and the table that follows

### Solving Applications of Systems of Linear Equations

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we will first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.
Step 1: Read the problem. Make sure all the words and ideas are understood.
Step 2: Identify what we are looking for.
Step 3: Name what we are looking for. Choose variables to represent those quantities.
Step 4: Translate into a system of equations.
Step 5: Solve the system of equations using good algebra techniques.
Step 6: Check the answer in the problem and make sure it makes sense.
Step 7: Answer the question with a complete sentence.

### Check Your Understanding

### Key Terms

1. system of linear equations
2. solutions of a system of equations
3. contradictions
4. identities
5. coincident lines
6. consistent system of linear equations
7. inconsistent system of linear equations

### Key Concepts

1. To solve a system of linear equations means finding the point or points where the two linear equations intersect.
2. Two lines can intersect at one point, no points if they are parallel, or every point if they are the same equation.
3. Systems of linear equations can be solved by graphing, by using substitution, or by using the elimination method.

### Videos

1. Practice with Solving Applications of Systems of Equations
2. Applications of Systems of Linear Equations


# Algebra

## Systems of Linear Inequalities in Two Variables

### Learning Objectives

After completing this section, you should be able to:
1. Demonstrate whether an ordered pair is a solution to a system of linear inequalities.
2. Solve systems of linear inequalities using graphical methods.
3. Graph systems of linear inequalities.
4. Interpret and solve applications of linear inequalities.
In this section, we will learn how to solve systems of linear inequalities in two variables. In Systems of Linear Equations in Two Variables, we learned how to solve for systems of linear equations in two variables and found a solution that would work in both equations. We can solve systems of inequalities by graphing each inequality (as discussed in Graphing Linear Equations and Inequalities) and putting these on the same coordinate system. The double-shaded part will be our solution to the system. There are many real-life examples for solving systems of linear inequalities.
Consider Ming who has two jobs to help her pay for college. She works at a local coffee shop for $7.50 per hour and at a research lab on campus for $12 per hour. Due to her busy class schedule, she cannot work more than 15 hours per week. If she needs to make at least $150 per week, can she work seven hours at the coffee shop and eight hours in the lab?

### Determining If an Ordered Pair Is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is 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 here.
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. The solution of a system of linear inequalities is shown as a shaded region in the -coordinate system that includes all the points whose ordered pairs make the inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

### Solving Systems of Linear Inequalities Using Graphical Methods

The solution to a single linear inequality was 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. We will review graphs of linear inequalities and solve the linear inequality from its graph.

### Graphing Systems of Linear Inequalities

We learned that the solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region by graphing, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.
Step 1: Graph the first inequality.
Graph the boundary line.
Shade in the side of the boundary line where the inequality is true.
Step 2: On the same grid, graph the second inequality.
Graph the boundary line.
Shade in the side of that boundary line where the inequality is true.
Step 3: The solution is the region where the shading overlaps.
Step 4: Check by choosing a test point.
Systems of linear inequalities where the boundary lines are parallel might have no solution. We will see this in the next example.
Some systems of linear inequalities where the boundary lines are parallel will have a solution. We will see this in the next example.

### Interpreting and Solving Applications of Linear Inequalities

When solving applications of systems of inequalities, first translate each condition into an inequality. Then 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 add inequalities to the system as additional requirements.

### Check Your Understanding

### Key Terms

1. system of linear inequalities

### Key Concepts

1. To solve a system of linear inequalities means to find the area(s) where the points in that area make all the linear inequalities true.
2. Systems of linear inequalities can be solved by graphing the linear equations associated with the inequalities, then 'testing' points to see whether the values of the point make the equation true or not.

### Videos

1. Solving Systems of Linear Inequalities by Graphing
2. Systems of Linear Inequalities


# Algebra

## Linear Programming

### Learning Objectives

After completing this section, you should be able to:
1. Compose an objective function to be minimized or maximized.
2. Compose inequalities representing a system application.
3. Apply linear programming to solve application problems.
Imagine you hear about some natural disaster striking a far-away country; it could be an earthquake, a fire, a tsunami, a tornado, a hurricane, or any other type of natural disaster. The survivors of this disaster need help—they especially need food, water, and medical supplies. You work for a company that has these supplies, and your company has decided to help by flying the needed supplies into the disaster area. They want to maximize the number of people they can help. However, there are practical constraints that need to be taken into consideration; the size of the airplanes, how much weight each airplane can carry, and so on. How do you solve this dilemma? This is where linear programming comes into play. Linear programming is a mathematical technique to solve problems involving finding maximums or minimums where a linear function is limited by various constraints.
As a field, linear programming began in the late 1930s and early 1940s. It was used by many countries during World War II; countries used linear programming to solve problems such as maximizing troop effectiveness, minimizing their own casualties, and maximizing the damage they could inflict upon the enemy. Later, businesses began to realize they could use the concept of linear programming to maximize output, minimize expenses, and so on. In short, linear programming is a method to solve problems that involve finding a maximum or minimum where a linear function is constrained by various factors.

### Compose an Objective Function to Be Minimized or Maximized

An objective function is a linear function in two or more variables that describes the quantity that needs to be maximized or minimized.

### Composing Inequalities Representing a System Application

For our two examples of profit and production, in an ideal world the profit a person makes and/or the number of products a company produces would have no restrictions. After all, who wouldn’t want to have an unrestricted profit? However in reality this is not the case; there are usually several variables that can restrict how much profit a person can make or how many products a company can produce. These restrictions are called constraints.
Many different variables can be constraints. When making or selling a product, the time available, the cost of manufacturing and the amount of raw materials are all constraints. In the opening scenario with the tsunami, the maximum weight on an airplane and the volume of cargo it can carry would be constraints. Constraints are expressed as linear inequalities; the list of constraints defined by the problem forms a system of linear inequalities that, along with the objective function, represent a system application.

### Applying Linear Programming to Solve Application Problems

There are four steps that need to be completed when solving a problem using linear programming. They are as follows:
Step 1: Compose an objective function to be minimized or maximized.
Step 2: Compose inequalities representing the constraints of the system.
Step 3: Graph the system of inequalities representing the constraints.
Step 4: Find the value of the objective function at each corner point of the graphed region.
The first two steps you have already learned. Let’s continue to use the same examples to illustrate Steps 3 and 4.

### Check Your Understanding

### Key Terms

1. linear programming
2. objective function
3. constraint

### Key Concepts

1. Linear programming is a mathematical technique to solve problems involving finding maximums or minimums where a linear function is limited by various constraints.
2. An objective function is a linear function in two or more variables that describes the quantity that needs to be maximized or minimized.
3. In linear programming, a constraint is a restriction that affects the maximum or minimum values of an objective function.
4. Through the creation of objective functions and restraints, a linear system can be developed and solved through linear programming.

### Projects

### Ratio and Proportion—Comparing Prices, Part 1

Go to your favorite coffee shops and find out what a same sized drink costs at each. You can do something similar for pizza as well. Find the unit rate (i.e., price per ounce or price per square inch). For example, go to your favorite coffee place and find the price per units on all their large coffee drinks. Or go to your favorite pizza place and compare prices of all their extra-large pizzas (by price per square inch). Write a report on the best deals.

### Ratio and Proportion—Comparing Prices, Part 2

Rather than comparing prices of different, but same sized drinks (or pizzas), compare unit prices of the same drinks but of different sizes. Find out what the best bargain is based on price per ounce, price per square inch, etc. For example, compare the prices of your favorite soft drink sold at a local store, but in various sizes (i.e., 12-ounce can, 16-ounce bottle, 20-ounce bottle, 1-liter bottle, and multipacks). Or go to a pizza place and find out what the best bargain is on their menu, based on price per square inch of pizza. Write a report on the best deals.

### Systems of Linear Inequalities—Comparing Cell Phone Plans

Go to the websites of different cell phone companies and compare their plans. Write a report on “the best deals. "Best Deals” doesn’t necessarily mean “cheapest.” You will need to look at what each company provides concerning restrictions (constraints) on minutes to talk. What are the constraints on the cell phone coverage for each company? Do they cover your area of the country well? Do they cover the entire United States well, or at least areas where you will be travelling? Is this coverage 5G, or is it less? Can you add a phone easily? Can you bring your previous phone number to this plan? The possibilities of constraints affecting each plan are several. So your task is to determine which plan is best, based on not only cost but also all constraints you deem important.

### Chapter Review

### Algebraic Expressions

### Linear Equations in One Variable with Applications

### Linear Inequalities in One Variable with Applications

### Ratio and Proportions

### Graphing Linear Equations and Inequalities

### Quadratic Equations with Two Variables with Applications

### Functions

### Graphing Functions

### System of Linear Equations in Two Variables

### Systems of Linear Inequalities in Two Variables

### Linear Programming

### Chapter Test


# Money Management

## Introduction

The topic of money management is a broad and sometimes complex one. Ultimately, personal money management involves managing both our debt and also our savings and investments.
In 2021, the average American had consumer debt balance of $96,371. Nearly $100,000 per person. And less than 25% of Americans are debt free. Consumer debt can include mortgages, credit cards, as well as student loans. A key question all consumers should consider is how to manage debt and not become overburdened by it. The first step is to create a budget, which puts earnings into perspective, indicating what we can, and cannot, afford. A budget also entails setting aside certain funds for savings and investment, which help us achieve our short- and long-term goals.
Creating a budget requires an understanding of how money—debt and savings—works. Initially, percentages and interest need to be understood. They drive most of what happens with debt and savings. With that understanding, discussions of buying a house, a car, or incurring credit card debt can be addressed from a financial perspective. All the while, retirement is waiting. Preparing for retirement involves saving and saving earlier rather than later. The power of compound interest is on full display when saving early.
This chapter covers some of the basics of money management: percentages, interest, budgeting, debt (student loans, mortgage, car, credit cards), savings, investments, and taxes.


# Money Management

## Understanding Percent

### Learning Objectives

After completing this section, you should be able to:
1. Define and calculate percent.
2. Convert between percent, decimal, and fractional values.
3. Calculate the total, percent, or part.
4. Solve application problems involving percents.
In 2020, the U.S. federal government budgeted $3.5 billion for the National Park Service, which appears to be a very large number (and is!) and a large portion of the total federal budget. However, the total outlays from the U.S. federal government in 2020 was $6.6 trillion. So, the amount budgeted for the National Park Service was less than one-tenth of 1 percent, or 1/10%, of the total outlays. This percent describes a specific number. Understanding that ratio puts the $3.5 billion budgeted to the National Park Service in perspective.
This chapter focuses on percent as a primary tool for understanding money management. The interest paid on debt, the interest earned through investments, and even taxes are entirely determined using percent. This section introduces the basics of working with this invaluable tool.

### Define and Calculate Percent

The word percent comes from the Latin phrase per centum, which means “by the hundred.” So any percent is a number divided by 100. Changing a percent to a fraction is to write the percent in its fractional form. To write % in its fractional form is to write the percent as the fraction .

### Convert Between Percent, Decimal, and Fractional Values

When any calculation with a percent is to be performed, the form of the percent must be changed, either to its fractional form or its decimal form. We can change a percent into decimal form by dividing the percent by 100 and representing the result as a decimal.
You should notice that, to convert from percent to decimal form, you can simply move the decimal two places to the left without performing the division.
You should notice that, to convert from decimal form to percent form, you can simply move the decimal two places to the right without performing the multiplication.

### Calculate the Total, Percent, or Part

The word “of” is used to indicate multiplication using fractions, as in “one-fourth of 56.” To find “one-fourth of 56” we would multiply 56 by one-fourth. We can think of percents as fractions with a specific denominator—100. So, to calculate “25% of 52,” we multiply 52 by 25%. But, first we need to convert the percent to either fractional form (25/100) or decimal form. Using the decimal form of 25% we have 0.25 × 52, which equals 13.
In this problem, 52 is the total or base, 25 is the percentage, and 13 is the percentage of 52, or the part of 52. This is sometimes referred to as the amount.
Knowing any two of the values in our formula allows us to calculate the third value. In the following example, we know the total and the percent, and are asked to find the percentage of the total.
In the previous example, we knew the total and the percent and found the part using our formula. We may instead know the percent and the part, but not the total. We can use our formula again to solve for the total.
Similarly, the percent can be found if the total and the percent of the total (the part) are known. This will result in the decimal form of the percent, so it must be converted to percent form.

### Solve Application Problems Involving Percents

Percents are frequently used in finance, research, science experiments, and even casual conversation. Understanding these types of values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percents will help you become an informed consumer of this information.
In most cases, working through what is presented requires you to identify that you are indeed working with a question of percents, which two of the three values that are related through percents are known, and which of the three values you need to find.

### Check Your Understanding

For any answer, round to two decimal places, if necessary.

### Key Terms

1. Percent
2. Fractional form
3. Decimal form
4. Total
5. Base
6. Percent of the total
7. Part
8. Amount

### Key Concepts

1. Know what a percent is as a fraction, a decimal, and as a part of the whole.
2. Use the percent equation to find any of the three values that are related by the equation.
3. Apply the percent equation in applications.

### Video

1. Finding Percent of a Total
2. Finding the Total from the Percent and the Part
3. Finding the Percent When the Total and the Part Are Known

### Formulas

part = percent x total


# Money Management

## Discounts, Markups, and Sales Tax

### Learning Objectives

After completing this section, you should be able to:
1. Calculate discounts.
2. Solve application problems involving discounts.
3. Calculate markups.
4. Solve application problems involving markups.
5. Compute sales tax.
6. Solve application problems involving sales tax.
Many people first encounter percentages during a retail transaction such as a percent discount (SALE! 25% off!!), or through sales tax ("Wait, I thought this was $1.99?"), a report that something has increased by some percentage of the previous value (NOW! 20% more!!). These are examples of percent decreases and percent increases. In this section, we discuss decrease, increase, and then the case of sales tax.

### Calculating Discounts

Retailers frequently hold sales to help move merchandise. The sale price is almost always expressed as some amount off the original price. These are discounts, a reduction in the price of something. The price after the discount is sometimes referred to as the reduced price or the sale price.
When a reduction is a percent discount, it is an application of percent, which was introduced in Understanding Percent. The formula used was . In a discount application, the discount plays the role of the part, the percent discount is the percentage, and the original price plays the role of the total.
When the original price and the percent discount are known, the discount and the sale price can be directly computed.
Sometimes the original price and the sale price of an item is known. From this, the percent discount can be computed using the formula , by solving for the percent discount.
Sometimes the sale price and the percent discount of an item are known. From this, the original price can be found. To avoid multiple steps, though, the formula that we will use is . The original price can be found by solving this equation for the original price.

### Solve Application Problems Involving Discounts

In application problems, identify what is given and what is to be found, using the terms that have been learned, such as discount, original price, percent discount, and sale price. Once you have identified those, use the appropriate formula (or formulas) to find the solution(s).

### Calculate Markups

When retailers purchase goods to sell, they pay a certain price, called the cost. The retailer then charges more than that amount for the goods. This increase is called the markup. This selling price, or retail price, is what the retailer charges the consumer in order to pay their own costs and make a profit. Markup, then is very similar to discount, except we add the markup, while we subtract the discount.
It should be noted that the formulas used for a markup are very similar to those for a discount, with addition replacing the subtraction.
Sometimes the cost and the retail price of an item are known. From this, the percent markup can be computed using the formula , by solving for the percent markup.
Sometimes the retail price and the percent markup of an item are known. From this, the cost can be found. To avoid multiple steps, though, the formula that we will use is . The cost can be found by solving this equation for the cost.

### Solve Application Problems Involving Markups

As before when working with application problems, be sure to look for what is given and identify what you are to find. Once you have evaluated the problem, use the appropriate formula to find the solution(s). These application problems address markups.

### Compute Sales Tax

Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. None of these sales tax revenues go to the federal government.
For example, North Carolina has a state sales tax of 4.75% while New Mexico has a state sales tax of 5%. Additionally, many counties in North Carolina charge an additional 2% sales tax, bringing the total sales tax for most (72 of the 100) counties in North Carolinians to 6.75%. However, in Durham, the county sales tax is 2.25% plus an additional 0.5% tax used to fund public transportation, bringing Durham County’s sales tax to 7%. To find the sales tax in a particular place, then, add other locality sales taxes to the base state sales tax rate.
How much we pay in sales tax depends on where we are, and what we are buying.
To determine the amount of sales tax on taxable purchase, we need to find the product of the purchase price, or marked price, and the sales tax rate for that locality.
You should notice that this the same as markup, except using a different term. Sales tax plays the role of markup, the purchase price plays the role of cost, and the tax rate plays the role of percent markup. This means all the strategies developed for markups apply to this situation, with the changes indicated.
As before, the information available might be different than only the purchase price and the sales tax rate. In these cases, use either  or  and solve for the indicated tax, price, or rate. These problems mirror those for percent markup.
Be aware, almost all sales tax rates are structured as full percentages, or half percent, or one-quarter percent, or three-quarter percent. This means the decimal value of the sales tax rate, written as a percent, will be either 0, as in 5.0%, 5 as in 7.5%, 25 as in 3.25%, or 75 as in 4.75%. When rounding for the sales tax percentage, be sure to use this guideline.

### Solve Application Problems Involving Sales Tax

Solving problems involving sales tax follows the same ideas and steps as solving problems for markups. But here we will use the following formula:
We can also use the formula:
This can be seen in the following examples.

### Check Your Understanding

### Key Terms

1. Discount
2. Cost
3. Markup
4. Retail price

### Key Concepts

1. Discounts are markdowns from an original price.
2. Mark-ups are increases to the price paid by a retailer to cover their costs.
3. be able to calculate the markup based on a percentage of the cost
4. Sales taxes vary from state to state and often county to county.
5. Retail prices, sales prices and percent discounts can be calculated if the other two values are known.
6. Original costs, retail prices, and percent markup can be calculated if the other two values are known.
7. In calculations, sales tax acts like a markup.

### Videos

1. Computing Price Based on a Percent Off Coupon
2. Finding Sales Tax Percentage

### Formulas