3 - Solving Basic Equations

Chapter 3

E

quations are an integral part of algebra. Since you will be coming

across equations at every turn,

understanding how to work with them is essential. Chapter 3 provides you with everything you need to know for solving basic equations.

Solving Basic

E

quations

In this Chapter... Solve an Equation with Addition and Subtraction Solve an Equation with Multiplication and Division Solve an Equation in Several Steps Solve for a Variable in an Equation with Multiple Variables Solve an Equation with Absolute Values Test Your Skills

Solve an Equation with Addition and Subtraction Addition and subtraction are the basic tools you will begin using to solve equations. Before you start adding and subtracting, however, you must understand the principle of keeping the equation balanced. Whenever you perform an operation on one side of the equals sign, you must perform the same operation on the other side so the equation will remain balanced and true. The object of your adding and subtracting is to remove or “cancel out” numbers. Canceling out numbers isolates the variable to one side of the equation, usually the left side, and is referred to How to Solve an Equation

as “solving for the variable.” For example, in the equation x + 7 = 9 you would subtract 7 from each side of the equals sign to cancel out the 7 originally found on the left side. In this example, we would have x + 7 – 7 = 9 – 7, which leaves an answer of x = 2. Thankfully, equations are easy to check. All you have to do is take the value you determined for the variable, replace the variable in the original equation with the value you found and then work out the problem. If one side equals the other, you have correctly solved for the variable. Solving Equations Example 1

x – 5 = 11

x + 7 = 20

x – 5 + 5 = 11 + 5

x + 7 – 7 = 20 – 7

x = 16

Check your answer

x – 5 = 11

13 + 7 = 20

11 = 11 Correct!

your goal is to place the variable on one side of the equation, usually the left side, and place all the numbers on the other side of the equation, usually the right side. This allows you to determine the value of the variable.

64

Check your answer

x + 7 = 20

16 – 5 = 11

• When solving equations,

x = 13

Note: A variable is a letter, such as x or y, which represents any number.

• To place a variable by

itself on one side of an equation, you can add or subtract the same number or variable on both sides of the equation and the equation will remain balanced.

20 = 20 Correct!

1 In this example, to place

the variable by itself on the left side of the equation, subtract 7 from both sides of the equation. This will remove, or cancel out, 7 from the left side of the equation.

2 To check your answer,

place the number you found into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

Chapter 3

Tip

How do I know when I should add or subtract to solve an equation? As a simple rule of thumb, you should do the opposite of what is done in the original equation. When a number is subtracted from a variable in an equation, such as x – 5 = 11, you add the subtracted number to both sides of the equation to cancel out the number and isolate the variable. If a number is added to a variable, such as x + 5 = 11, you can subtract the added number from both sides of the equation to cancel out the number and isolate the variable.

Solving Equations Example 2

5 = 20 + x 5 – 20 = 20 + x – 20 –15 = x

x = –15

Check your answer

5 = 20 + x 5 = 20 + (–15) 5 = 5 Correct!

1 In this example, to

place the variable by itself on the right side of the equation, subtract 20 from both sides of the equation.

2 To check your answer, place the number you found into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

ctice Pra

Solving Basic Equations

Solve for the variable in each of the following equations. You can check your answers on page 252.

1) x + 2 = 4 2) x – 3 = 10 3) 5 = x – 4 4) –3 = x – 3 5) 2 x – 3 = x + 7 6) 5 x + 1 = 6 x – 4

Solving Equations Example 3

8x – 6 = 7x + 3 8x – 6 + 6 = 7x + 3 + 6 8x = 7x + 9 8x – 7x = 7x + 9 – 7x

x=9 Check your answer

8x – 6 = 7x + 3 8(9) – 6 = 7(9) + 3 72 – 6 = 63 + 3 66 = 66 Correct!

1 In this example, to

place all the numbers on the right side of the equation, add 6 to both sides of the equation. You can then subtract 7 x from both sides of the equation to move all the variables to the left side of the equation.

2 To check your answer,

place the number you found into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

65

Solve an Equation with Multiplication and Division When multiplying and dividing in equations, you must remember the principle of balance, just like when you add and subtract within equations. If you perform an operation on one side of the equals sign, you have to perform the same operation on the other side in order for the equation to remain balanced and true. Your aim in multiplying and dividing in an equation is to remove or “cancel out” numbers and isolate the variable to one side of the equation, usually the left side. This is called “solving for the variable.” How to Solve an Equation

5 x = 20

8 x = 24

5 x = 20 5 5 x=4

8 x = 24 8 8 x=3

5 x = 20 5(4) = 20 20 = 20

When solving an equation, your goal is to place the variable by itself on one side of the equation, usually the left side, and place all the numbers on the other side of the equation, usually the right side. This allows you to determine the value of the variable.

66

To check your answers, take the value you determined for the variable, substitute it into the original equation and then work out the problem. If one side equals the other, you have correctly solved for the variable. Solving Equations Example 1

Check your answer

•

When working with equations, you will often have to remove a coefficient, which is a number next to a variable, such as 4a. Since the variable is being multiplied by the coefficient, you should divide both sides by the coefficient to cancel out the coefficient and isolate the variable.

Correct!

• To place a variable by

itself on one side of an equation, you can multiply or divide by the same number on both sides of the equation and the equation will remain balanced.

Check your answer

8 x = 24 8(3) = 24 24 = 24 Correct!

1 In this example, to place the variable by itself on the left side of the equation, divide both sides of the equation by the number in front of the variable, called the coefficient. This will leave the x variable by itself on the left side of the equation because 88 equals 1.

2 To check your answer,

place the number you found into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

Chapter 3

Tip

How do I know when I should multiply or divide to solve an equation? Generally, you should do the opposite of what is done in the original equation. When a variable is divided by a number in an equation, such as 5x = 10, you can multiply by the same number on both sides of the equation to cancel out the number and isolate the variable. If a variable is multiplied by a number, such as 5x = 20, you can divide both sides of the equation by the same number to isolate the variable.

Solving Equations Example 2

ctice Pra

Solving Basic Equations

Solve for the variable in each of the following equations. You can check your answers on page 252.

x

1) 2 = 1 2) 4 x = 16 3) –3 x = 9 4) x = –5 2 x 5) =3 7 2 x 6) = 5 3 4 Solving Equations Example 3

x = 30 2 x x 2 = 30 2 x = 60

x

2

Check your answer

x = 30 2 60 = 30 2 30 = 30 Correct!

2x = 6 5 2x x 5 = 6x 5 5 2 2 30 x= = 15 2 Check your answer

2x = 6 5 2 (15) = 6 5 30 = 6 5 6=6

1 In this example, to

place the variable by itself on the left side of the equation, you must multiply both sides of the equation by the number divided into the variable.

2 To check your answer,

place the number you found into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

1 In this example, to place the

variable by itself on the left side of the equation, multiply both sides of the equation by the reciprocal of the fraction in front of the variable. Note: To determine the reciprocal of a fraction, switch the top and bottom numbers in the fraction. For example, 25 becomes 52 . When you multiply a fraction by its reciprocal, the result is 1 .

Correct!

2 To check your

answer, place the number you found into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

67

Solve an Equation in Several Steps Even when you are working with large, complex problems, your goal should be to isolate the variable to one side of the equation and find the value of the variable. Three easy steps, using skills you have learned in the previous pages, will guide you when you are confronted with a complicated equation. The first step in solving complex equations is to simplify the equation as much as you can on either side of the equals sign. You will find the distributive property particularly useful, as it eliminates those pesky parentheses. The second step involves using

addition and subtraction on both sides of the equation to isolate the variable on one side of the equation. For the third step, you multiply and divide on both sides of the equation to determine the value of the variable. Finally, don’t forget the unofficial fourth step: check your answer! Place the number you determined for the variable into the original equation and solve the problem. If both sides of the equation are equal, you have correctly solved the equation.

Step 1: Simplify Each Side of the Equation

6 x + 4(2 x + 3) = 34 + 3 x

6 x + 8 x + 12 = 34 + 3 x

6 x + 8 x + 12 = 34 + 3 x

14 x + 12 = 34 + 3 x

1 If an equation contains

parentheses ( ), use the distributive property on each side of the equals sign ( = ) to remove the parentheses that surround numbers and variables.

• In this example, 4(2x

+ 3) works out to 8x + 1 2 .

68

Note: The distributive property allows you to remove a set of parentheses by multiplying each number or variable inside the parentheses by a number or variable directly outside the parentheses. For more information on the distributive property, see page 30.

2 Add, subtract, multiply

and divide any numbers or variables that can be combined on the same side of the equals sign ( = ) .

• In this example, 6x equals 1 4x.

+ 8x

Note: For information on adding and subtracting variables, see page 36. For information on multiplying and dividing variables, see page 38.

Chapter 3

Tip

What should I do if I end up with a negative variable? If you end up with a negative sign (–) in front of a variable, such as –y = 14, you will have to make the variable positive to finish solving the equation. Fortunately, this is a simple task—just multiply both sides of the equation by –1. The equation remains balanced and a negative sign (–) will no longer appear in front of the variable. In the case of –y = 14, you would come out with a solution of y = –14.

Step 2: Add and Subtract

ctice Pra

Solving Basic Equations

Solve for the variable in each of the following equations. You can check your answers on page 252.

1) 3 x = 6 2) 4 x – 2 = 14 3) 2 x + 10 = 3 x – 5 4) x + 5 – 2 x + 3 = 7 – 3 x – 1 5) 2( x – 1) = 3( x + 1) – 3 6) 5(2 x – 3) + x = 2(2 – x ) + 7

Step 3: Multiply and Divide

11 x = 22 11 x = 22 11 11 x=2

14 x + 12 = 34 + 3 x 14 x + 12 – 12 = 34 + 3 x – 12 14 x = 22 + 3 x 14 x – 3 x = 22 + 3 x – 3 x 11 x = 22

Check your answer

6 x + 4(2 x + 3) = 34 + 3 x 6(2) + 4(2

x

2 + 3) = 34 + 3(2)

12 + 4(7) = 34 + 6 40 = 40

3 Determine which

numbers and variables you need to add or subtract to place the variable by itself on one side of the equation, usually the left side. Then add or subtract the same numbers and variables on both sides of the equation.

• In this example, subtract 12

from both sides of the equation to move all the numbers to the right side of the equation. You can then subtract 3x from both sides of the equation to place all the variables on the left side of the equation.

Note: For more information on adding and subtracting numbers and variables in equations, see page 64.

4 Determine which

numbers you need to multiply or divide by to place the variable by itself on one side of the equation. Then multiply or divide by the same numbers on both sides of the equation.

Correct!

• In this example, divide

both sides of the equation by 11 to place the variable by itself on the left side of the equation. The variable x remains on the left side of the equation by itself because 11 ÷ 11 equals 1. Note: For more information on multiplying and dividing numbers in equations, see page 66.

69

Solve for a Variable in an Equation with Multiple Variables In algebra, it is common to come across equations that have more than one variable. An equation containing multiple variables, such as a – 2 = 4 + b, shows you the relationship between the variables. You can easily isolate any of the variables to one side of the equation to simplify the relationship. For example, if you know that a = 6 + b, you do not need to know the numerical value of a in

order to define b. By simply rearranging the equation, you can find that b = a – 6. Working with equations containing multiple variables is really no more difficult than working with equations that have just one variable. The only difference is that your final result will have variables on both sides of the equation.

Step 1: Simplify Each Side of the Equation

Solve for

x in the equation

2(3 x + 6) + 4 x – y = 32 6 x + 12 + 4 x – y = 32 10 x + 12 – y = 32

2(3 x + 6) + 4 x – y = 32 6 x + 12 + 4 x – y = 32

1 If an equation contains

parentheses ( ), use the distributive property on each side of the equals sign ( = ) to remove the parentheses that surround numbers and variables.

• In this example, 2(3x

+ 6) works out to 6x + 1 2.

70

Note: The distributive property allows you to remove a set of parentheses by multiplying each number or variable inside the parentheses by a number or variable directly outside the parentheses. For more information on the distributive property, see page 30.

2 Add, subtract, multiply

and divide any numbers or variables that can be combined on each side of the equation.

• In this example, 6x + 4 x equals 1 0 x.

Note: For information on adding and subtracting variables, see page 36. For information on multiplying and dividing variables, see page 38.

Chapter 3

Tip

Once I have solved for a variable in an equation, do I need to further simplify the answer? If you end up with a fraction that has two or more numbers and variables added or subtracted in the numerator, you can simplify your answer by splitting the fraction into smaller fractions. Each new fraction will contain one term of the original numerator and a duplicate of the denominator.

y y 20 + y = 20 + = 2+ x = 10 10 10 10

Step 2: Add and Subtract

ctice Pra

Solving Basic Equations

Solve for x in the following equations. You can check your answers on page 252.

1. 2 x + 3 – y = 5 2. x + 6 y = 4 – x 3. 3( y + 4) = 5 + x Solve for y in the following equations. You can check your answers on page 252.

4. 2( x – 1) + 2 y = 5 + y 5. 3( x + y ) – 4 y = 2 6. y + x + 2 = 4 – y + x

Step 3: Multiply and Divide

10 x + 12 – y = 32

10 x = 20 + y

10 x + 12 – y – 12 = 32 – 12

20 + y 10 x = 10 10 20 + y x= 10

10 x – y = 20 10 x – y + y = 20 + y 10 x = 20 + y

3 Determine which numbers • In this example,

5 Determine which

4 Add or subtract the

6 Multiply or divide by

and variables you need to add or subtract to place the variable by itself on one side of the equation, usually the left side. numbers and/or variables you determined in step 3 on both sides of the equation.

subtract 12 from both sides of the equation. You can then add y to both sides of the equation.

Note: For information on adding and subtracting numbers and variables in equations, see page 64.

numbers you need to multiply or divide by to place the variable by itself on one side of the equation. the numbers you determined in step 5 on both sides of the equation.

• In this example, divide

both sides of the equation by 1 0. The x variable remains by itself on the left side of the equation because 1 0 ÷ 10 equals 1. Note: For information on multiplying and dividing numbers in equations, see page 66.

71

Solve an Equation with Absolute Values Every once in a while you will come across an equation that contains an innocent-looking pair of absolute value symbols ( | | ) such as x = | y + 3| . These equations are known as absolute value equations and they are not as simple as they look. Remember that the absolute value of a number or variable is always the positive value of that number or variable, whether the number or variable is positive or negative. So |a| can equal either a or –a.

You will have to approach absolute value equations slightly differently than you would approach a normal equation because an absolute value equation may result in more than one answer. For example, the absolute value equation |x| = 1 + 3 can be simplified as |x| = 4. You can then determine that x = 4 or –4. You get two answers for the price of one!

Step 1: Isolate the Absolute Value Expression

5 | 4 x + 2 | + 6 = 56 5 | 4 x + 2 | + 6 = 56

5 | 4 x + 2 | + 6 – 6 = 56 – 6 5 | 4 x + 2 | = 50

• When solving an

equation that contains an absolute value expression, you first need to place the absolute value expression by itself on one side of the equation.

72

Note: An absolute value expression consists of variables and/or numbers inside absolute value symbols | | . The result of an absolute value expression is always greater than zero or equal to zero. For more information on absolute values, see page 24.

1 Determine which numbers

and variables you need to add or subtract to place the absolute value expression by itself on one side of the equation.

2 Add or subtract the

numbers and variables you determined in step 1 on both sides of the equation.

• In this example,

subtract 6 from both sides of the equation to remove 6 from the left side of the equation.

Note: For information on adding and subtracting numbers and variables in equations, see page 64.

Chapter 3

Tip

How do I solve a simple equation with absolute values? Simple equations containing absolute values, like | x | = 8, are fairly simple to evaluate. Remember that all you are being asked for is the possible solutions for x. In the example | x | = 8, both 8 and –8 have an absolute value of 8, so x = 8 or –8.

Tip

Solving Basic Equations

Why do I need to create two equations from each equation that contains absolute values? The key to solving an equation that contains absolute values is removing the absolute value symbols ( | | ). Values within these symbols may have a negative or positive value, so you have to work out both possibilities by creating and solving two equations.

Step 2: Create Two New Equations

5 | 4 x + 2 | = 50

| 4 x + 2 | = 10 5 | 4 x + 2 | = 50 5 5

4 x + 2 = 10

4 x + 2 = –10

| 4 x + 2 | = 10

3 Determine which

numbers you need to multiply or divide by to place the absolute value expression by itself on one side of the equation.

4 Multiply or divide

by the numbers you determined in step 3 on both sides of the equation.

• In this example,

divide both sides of the equation by 5 to remove 5 from the left side of the equation. Note: For information on multiplying and dividing numbers in equations, see page 66.

• You now need to create

two new equations from the original equation.

5 Create two new

equations that look like the original equation, but without the absolute value symbols | | .

6 For the second

equation, place a negative sign (–) in front of the value on the opposite side of the absolute value expression.

CONTINUED

73

Solve an Equation with Absolute Values continued After you have created two new equations from the original absolute value equation, you can solve each of the two new equations and determine the two possible answers for the variable in an absolute value equation. The fun doesn’t end there, though. Even though it is twice the work, it is vital to check both of the numbers you found for the variable. When you place a number you found for the variable into the original equation and

work out the equation, both sides of the equation must equal each other in order to have a solution. Sometimes both of the numbers you found for the variable will solve the equation. In other instances, one or both of your numbers will fail to solve the equation. This means that either you have made a mistake and should double-check your calculations or there is only one or no solution for the equation.

Step 3: Solve the New Equations

4 x + 2 = 10

4 x + 2 = –10

4 x + 2 – 2 = 10 – 2

4 x + 2 – 2 = –10 – 2

4x = 8

4 x = –12

• You now need to solve

8 Add or subtract the

7 In both equations,

• In both examples,

the two new equations separately in order to determine the answer for the original equation. determine which numbers and variables you need to add or subtract to place the variable by itself on one side of the equation.

74

numbers and/or variables you determined in step 7 on both sides of the equation. subtract 2 from both sides of the equation.

4x = 8

4x = –12

4x = 8 4 4

4 x = –12 4 4

x=2

x = –3

9 In both equations,

determine which numbers you need to multiply or divide by to place the variable by itself on one side of the equation.

10 Multiply or divide

by the numbers you determined in step 9 on both sides of the equation.

• In both examples,

divide by 4 on both sides of the equation.

Chapter 3

Tip

Is there a solution to every equation with absolute values? No. As you work with equations that contain absolute values, you will sometimes come across equations that have no solution. One dead give-away is when the absolute value equals a negative number, as in | 4x + 2 | = –10. Don’t even bother working out these equations because no matter what x equals, the absolute value of the expression within the absolute value symbols cannot equal a negative number.

ctice Pra

Solving Basic Equations

Solve the following equations containing absolute values. You can check your answers on page 252.

1) | x | = 5 2) | 2 x – 3 | = 5 3) 2 | x – 1 | = 8 4) 2 | 3 x – 4 | – 3 = 5 5) | 3 x – 9 | = 0 6) 3 | 2 x + 6 | + 5 = 2

Step 4: Check Your Answers

If x = 2

If

x = –3

5 | 4 x + 2 | + 6 = 56

5 | 4 x + 2 | + 6 = 56

5 | 4(2) + 2 | + 6 = 56

5 | 4(–3) + 2 | + 6 = 56

5 | 8 + 2 | + 6 = 56

5 | –12 + 2 | + 6 = 56

5 | 10 | + 6 = 56

5 | –10 | + 6 = 56

5

x

10 + 6 = 56 56 = 56 Correct!

11 To check your answers, place

the first number you found into the original equation and solve the problem. If both sides of the equation are equal, this number correctly solves the equation.

5

x

10 + 6 = 56 56 = 56 Correct!

12 Place the second number you

found into the original equation and solve the problem. If both sides of the equation are equal, this number also correctly solves the equation.

75

Test Your Skills

Solving Basic Equations Question 1. Solve the following equations. a) 2 + x = 5

g) –8 x = –5

b) x – 4 = 10

h) x = 6 4 5 i) x = 10 3 j) 2 x = 1

c) x + 3 = 0 d) 2 x = –8 e) 3 x = 5 f) – x = –2

k) 3 x = 3 7

Question 2. Solve the following equations. a) 2 x + x + 1 = 2 b) 2 x + 1 = x c) 2 x + 3 = 4 x + 4 d) 20 – x (4 + 1) = 5 x e) 8 x – 2(5 x + 6) = 20 f) 5 x + 4(2 x – 7) = –15

Question 3. Solve for y in the following equations. a) y + x = 0 b) x + y = 2 + x c) 2 y + 3 x = 5 – x d) x – 1 = y + 4 e) y + x + y + 1 = 2 x – 3 y + 8 f) 2( x – y ) = 3( x + y + 1)

76

Chapter 3

Solving Basic Equations

Question 4. Solve the following equations. a) b) c) d) e) f) g)

|x | = 3 |–x | = 2 |x | = 0 | x | = –4 |x – 2| = 1 | x – 10 | = 3 |x + 5| = 5

Question 5. Solve the following equations. a) b) c) d) e) f) g) h)

|2x | = 4 |3x | = 8 | 2 x – 1 | = –1 |3x – 6| = 0 |2x – 1| = 1 |5x – 2| + 4 = 6 2 | 3 x – 1 | = 20 4 | 2 x + 3 | + 4 = 20

You can check your answers on pages 268-269.

77