3 - Solving Basic Equations

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Chapter 3

Equations are an integralpart of algebra. Sinceyou will be coming

across equations at every turn,understanding how to work withthem is essential. Chapter 3provides you with everythingyou need to know for solvingbasic equations.



Solve an Equation with Addition and Subtraction

64

Solving Equations Example 1How to Solve an Equation

x – 5 = 11

x – 5 + 5 = 11 + 5

x = 16

Check your answer

x – 5 = 11

1 6 – 5 = 11

11 = 11 Correct!

When solving equations,

your goal is to place thevariable on one side ofthe equation, usuallythe left side, and placeall the numbers on theother side of theequation, usua lly theright side. This allowsyou to determine thevalue of the variable.

Note: A variable is a letter,

such as x or y , whichrepresents any number.

To place a variable byitself on one side ofan equa tion, you canadd or subtract thesame number orvariable on both sidesof the equation andthe equation will

remain balanced.

In this example, to place

the variable by itself on theleft side of the equation,subtract 7 from both sidesof the equation. This willremove, or cancel out, 7

from the left side of theequation.

To check your answer,

place the number youfound into the originalequation and solve theproblem. If both sidesof the equation areequal, you havecorrectly solved theequation.

•

•

1 2

Addition and subtraction are the basic tools youwill begin using to solve equations. Before youstart adding and subtracting, however, you mustunderstand the principle of keeping the equationbalanced. Whenever you perform an operationon one side of the equals sign, you must performthe same operation on the other side so theequation will remain balanced and true.

The object of your adding and subtracting is toremove or “cancel out” numbers. Canceling outnumbers isolates the variable to one side of theequation, usually the left side, and is referred to

as “solving for the variable.” For example, inthe equation x + 7 = 9 you would subtract 7from each side of the equals sign to cancel outthe 7 originally found on the left side. In thisexample, we would have x + 7 – 7 = 9 – 7, whichleaves an answer of x = 2.

Thankfully, equations are easy to check. All you

have to do is take the value you determined forthe variable, replace the variable in the originalequation with the value you found and thenwork out the problem. If one side equals theother, you have correctly solved for the variable.

x + 7 = 20

x + 7 – 7 = 20 – 7

x = 13

Check your answer

x + 7 = 20

1 3 + 7 = 20

20 = 20 Correct!



Solving Basic EquationsChapter 3

65

Solving Equations Example 2 Solving Equations Example 3

P r a c t i c e Tip

Solve for the variable in each of the following equations. You can

check your answers on page 252.1) x + 2 = 42) x – 3 = 103) 5 = x – 4

4) –3 = x – 35) 2x – 3 = x + 76)

5x

+ 1 = 6x

– 4

How do I know when I should add or subtract to solve an equation?

As a simple rule of thumb, you shoulddo the opposite of what is done inthe original equation. When a numberis subtracted from a variable in anequation, such as x – 5 = 11, you add thesubtracted number to both sides of theequation to cancel out the number andisolate the variable. If a number is addedto a variable, such as x + 5 = 11, you can

subtract the added number from bothsides of the equation to cancel out thenumber and isolate the variable.

In this example, to

place the variable byitself on the rightside of the equation,subtract 20 fromboth sides of theequation.

In this example, to

place all the numberson the right side of theequation, add 6 to bothsides of the equation.You can then subtract7 x from both sides ofthe equation to moveall the variables to theleft side of theequation.


place the number youfound into the originalequation and solve theproblem. If both sidesof the equation areequal, you havecorrect ly solved theequation.

To check your answer, place

the number you found intothe original equation andsolve the problem. If bothsides of the equation areequal, you have correctlysolved the equation.

1 2 1 2

5 = 20 + x

5 – 20 = 20 + x – 20

–15 = x

x = –15

Check your answer

5 = 2 0 + x

5 = 20 + (–15)

5 = 5 Correct!

8x – 6 = 7 x + 3

8x – 6 + 6 = 7 x + 3 + 6

8 x = 7 x + 9

8x – 7 x = 7 x + 9 – 7 x

x = 9

Check your answer

8x – 6 = 7 x + 38(9) – 6 = 7(9) + 3

72 – 6 = 63 + 3

6 6 = 66 Correct!



Solve an Equation withMultiplication and Division

66

Solving Equations Example 1How to Solve an Equation

When solving an

equation, your goal isto place the variable byitself on one side of theequation, usually theleft side, and placeall the numbers onthe other side of theequation, usually theright side. This allowsyou to determine the

value of the variable.

To place a variable by

itself on one side ofan equation, you canmultiply or divide by thesame number on bothsides of the equationand the equation willremain balanced.

In this example, to place

the variable by itself onthe l eft side of theequation, divide bothsides of the equation bythe number in front ofthe variable, called thecoefficient. This willleave the x variable byitself on the left side ofthe equation because

equals 1 .



• • 1 2

When multiplying and dividing in equations, youmust remember the principle of balance, just likewhen you add and subtract within equations. Ifyou perform an operation on one side of the equalssign, you have to perform the same operation onthe other side in order for the equation to remainbalanced and true.

Your aim in multiplying and dividing in an equationis to remove or “cancel out” numbers and isolatethe variable to one side of the equation, usually theleft side. This is called “solving for the variable.”

When working with equations, you will oftenhave to remove a coefficient, which is a numbernext to a variable, such as 4 a . Since the variableis being multiplied by the coefficient, you shoulddivide both sides by the coefficient to cancel outthe coefficient and isolate the variable.

To check your answers, take the value you

determined for the variable, substitute it intothe original equation and then work out theproblem. If one side equals the other, you havecorrectly solved for the variable.

88

5x = 20=

x = 4

205

5x

5

Check your answer

5x = 205(4) = 20

20 = 20 Correct!

8x = 24=

x = 3

248

8x

8

Check your answer

8x = 248(3) = 24

24 = 24 Correct!




67

Solving Equations Example 2 Solving Equations Example 3

P r a c t i c e Tip

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

1) = 12) 4x = 163) –3 x = 9

4) = –5

5) = 36) =

How do I know when I should multiply or divide to solve an equation? Generally, you should do the oppositeof what is done in the original equation.When a variable is divided by a numberin an equation, such as = 10, you canmultiply by the same number on bothsides of the equation to cancel out thenumber and isolate the variable. If avariable is multiplied by a number, such

as 5 x = 20 , you can divide both sidesof the equation by the same numberto isolate the variable.

x2

x2x72 x3

54

In this example, to

place the variable byitself on the left sideof the equation, youmust multiply bothsides of the equationby the number dividedinto the variable.

In this example, to place the

variable by itself on the leftside of the equation, multiplyboth sides of the equation bythe reciprocal of the fractionin front of the variable.Note: To determine the reciprocalof a fraction, switch the top andbottom numbers in the fraction.For example, becomes .When you multiply a fraction by

its reciprocal, the result is 1 .

To check your

answer, place thenumber you foundinto the originalequation and solvethe problem. Ifboth sides of theequation are equal,you have correctlysolved theequation.



1 2 1 2

25

52

= 30

x 2 = 30 x 2

x = 60

x

2

Check your answer

= 30

= 30

30 = 30 Correct!

x

2602

x

2

x = 6

x x = 6 x

x = = 15302

25

Check your answer

x = 6

(15 ) = 6

= 6

6 = 6 Correct!

305

25

25

52

52

25

x 5



Solve an Equationin Several Steps

68

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

14 x + 12 = 34 + 3 x

Step 1: Simplify Each Side of the Equation

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

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

If an equation containsparentheses ( ) , use thedistributive property oneach side of the equalssign (= ) to remove theparentheses that surroundnumbers and variables.In this example, 4(2 x + 3)works out to 8 x + 12 .

Note: The distributiveproperty allows youto remove a setof parentheses bymultiplying each numberor variable inside theparentheses by a numberor variable directly outsidethe parentheses. Formore information onthe distributive property,see page 30.

Add, subtract, multiplyand divide any numbers orvariables that can becombined on the sameside of the equals sign (= ) .In this example, 6 x + 8 x

equals 14 x .

Note: For information onadding and subtractingvariables, see page 36.For information onmultiplying and dividingvariables, see page 38.

•

•

21

Even when you are working with large, complexproblems, your goal should be to isolate the variableto one side of the equation and find the value ofthe variable. Three easy steps, using skills you havelearned in the previous pages, will guide you whenyou are confronted with a complicated equation.

The first step in solving complex equations is tosimplify the equation as much as you can on eitherside of th e equals sign. You will find the distributiveproperty particularly useful, as it eliminates thosepesky parentheses. The second step involves using

addition and subtraction on both sides of theequation to isolate the variable on one side ofthe equation. For the third step, you multiplyand divide on both sides of the equation todetermine the value of the variable.

Finally, don’t forget the unofficial fourth step:check your answer! Place the number youdetermined for the variable into the originalequation and solve the problem. If both sidesof the equation are equal, you have correctlysolved the equation.




69

Step 2: Add and Subtract Step 3: Multiply and Divide

11 x = 22

=

x = 2

2211

11 x

11

Check your answer

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

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

12 + 4(7) = 34 + 6

40 = 40 Correct!

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

P r a c t i c e Tip

Solve for the variable in each of the

following equ ations. You can check your answers on page 252.

1) 3x = 62) 4x – 2 = 143) 2x + 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

What should I do if I end up with a

negative variab le? If you end up with a negative sign (–) infront of a variable, such as – y = 14, youwill have to make the variable positive tofinish solving the equation. Fortunately,this is a simple task—just multiply bothsides of the equation by –1. The equationremains balanced and a negative sign (–)will no longer appear in front of thevariable. In the case of – y = 14, you wouldcome out with a solution of y = –14.

• •Determine whichnumbers andvariables you needto add or subtract toplace the variable byitself on one side ofthe equation, usuallythe left side. Thenadd or subtract thesame numbers andvariables on bothsides of the equation.

Determine whichnumbers you need tomultiply or divide byto place the variableby itself on one sideof the equation.Then multiply ordivide by the samenumbers on bothsides of the equation.

In this example, divideboth sides of the equationby 11 to place the variableby itself on the left side ofthe equation. The variable x

remains on the left sideof the equation by itselfbecause 11 ÷ 11 equals 1 .Note: For more informationon multiplying and dividingnumbers in equations, see

page 66.

In this example, subtract 12from both sides of theequation to move all thenumbers to the right side ofthe equation. Yo u can thensubtract 3 x from both sidesof the equation to place allthe variables on the left sideof the equation.Note: For more information onadding and subtracting numbers and

variables in equations, see page 64.

3 4



Solve for a Variable in anEquation with Multiple Variables

70

Step 1: Simplify Each Side of the Equation

Solve for x in the equation

2(3 x + 6) + 4 x – y = 32

2(3 x + 6) + 4 x – y = 32

6x + 12 + 4 x – y = 32

If an equation containsparenthese s ( ) , use thedistributive property oneach side of the equalssign (= ) to remove theparentheses that surroundnumbers and variables.In this example, 2(3 x + 6)works out to 6 x + 12 .

Note: The distributiveproperty allows youto remove a setof parentheses bymultiplying each numberor variable inside theparentheses by a numberor variable directlyoutside the parentheses.For more information onthe distributive property,see page 30.

Add, subtract, multiplyand divide any numbersor variables that can becombined on each sideof the equation.In this example, 6 x + 4 xequals 10 x .

Note: For information onadding and subtractingvariables, see page 36.For information onmultiplying and dividingvariables, see page 38.

•

•

21

In algebra, it is common to come across equationsthat have more than one variable. An equationcontaining multiple variables, such as a – 2 = 4 + b ,shows you the relationship between the variables.You can easily isolate any of the variables to oneside of the equation to simplify the relationship.

For example, if you know that a = 6 + b , you donot need to know the numerical value of a in

order to define b . By simply rearranging theequation, you can find that b = a – 6.

Working with equations containing multiplevariables is really no more difficult than workingwith equations that have just one variable. Theonly difference is that your final result will havevariables on both sides of the equation.

6x + 12 + 4x – y = 32

10 x + 12 – y = 32




71

Step 2: Add and Subtract Step 3: Multiply and Divide

P r a c t i c e Tip

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

1. 2x + 3 – y = 52. 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 = 26. y + x + 2 = 4 – y + x

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 hastwo or more numbers and variables addedor subtracted in the numerator, you cansimplify your answer by splitting the fractioninto smaller fractions. Each new fraction willcontain one term of the original numeratorand a duplicate of the denominator.

• •Determine which numbersand variables you need toadd or subtract to placethe variable by itself onone side of the equation,usually the left side.Add or subtract thenumbers and/or variablesyou determined in step 3

on both sides of theequation.

Determine whichnumbers you need tomultiply or di vide byto place the variableby itself on one sideof the equation.Multiply or divide bythe numbers youdetermined in step 5

on both sides of theequation.

In this example, divideboth sides of theequation by 10 . The xvariable remains byitself on the left side ofthe equation because10 ÷ 10 equals 1 .Note: For information onmultiplying and dividingnumbers in equations,see page 66.

In this example,subtract 12 from bothsides of the equation.You can then add y toboth sides of theequation.Note: For information onadding and subtractingnumbers and variables inequations, see page 64.

3

4

5

6

10 x + 12 – y = 32

10 x + 12 – y – 12 = 32 – 12

10 x – y = 20

10 x – y + y = 20 + y

10 x = 20 + y

10 x = 20 + y

=

x =

20 + y10

10 x10

x = = + = 2 + y10 y10201020 + y10

20 + y10



Solve an Equation with Absolute Values

72

Step 1: Isolate the Absolute Value Expression

5 | 4 x + 2 | + 6 = 56

When solving anequation thatconta ins an absolutevalue expression, youfirst need to placethe absolute valueexpression by itselfon one si de of theequation.

Note: An absolute valueexpression consists of variablesand/or numbers inside absolutevalue symbols | |. The result ofan absolute value expression isalways greater than zero or equalto zero. For more information onabsolute values, see page 24.

Determine which numbersand variables you need toad d or subtract to place theabsolute value expressionby itself on one side of theequation.Add or subtract thenumbers and variables youdetermined in step 1 onboth sides of the equation.

In this example,subtract 6 from bothsides of the equationto remove 6 from theleft side of th eequation.Note: For information onadding and subtractingnumbers and variables inequations, see page 64.

• •1

2

Every once in a while you will come across anequation that contains an innocent-looking pairof absolute value symbols ( | | ) such as x = | y + 3 | .These equations are known as absolute valueequations and they are not as simple as they look.

Remember that the absolute value of a numberor variable is always the positive value of thatnumber or variable, whether the number orvariable is positive or negative. So | a| canequal either a or – a .

You will have to approach absolute valueequations slightly differently than you wouldapproach a normal equation because anabsolute value equation may result in morethan one answer. For example, the absolutevalue equation | x | = 1 + 3 can be simplifiedas | x | = 4. You can then determine that x = 4

or –4. You get two answers for the price of one!

5 | 4 x + 2 | + 6 = 56

5 | 4 x + 2 | + 6 – 6 = 56 – 6

5 | 4 x + 2 | = 50




73

Step 2: Create Two New Equations

Tip Tip

Why do I need to create two equations from each equation that contains absolute values?

The key to solving an equation thatcontains absolu te values is removingthe absolute value symbols ( | | ).Values within these symbols may havea negative or positive value, so youhave to work out both possibilities bycreati ng and solving two equations.

How do I solve a simple equation with absolute values?

Simple equations containingabsolute values, like | x | = 8 ,are fairly simple to evaluate.Remember that all you are beingasked for is the possible solutionsfor x . In the example | x | = 8,both 8 and –8 have an absolutevalue of 8, so x = 8 or –8.

••Determine whichnumbers you need tomultiply or divide byto place the absolutevalue expression byitself on one side ofthe equation.Multiply or divideby the numbers youdetermined in step 3

on both sides of the

equation.

You now need to createtwo new equations fromthe original equation.Create two newequations that look likethe original equation,but without the absolutevalue symbols | |.

For the secondequation, place anegative sign ( – ) infront of the value onthe opposite side ofthe absolute valueexpression.

In this example,divide both sides ofthe equation by 5 toremove 5 from the leftside of the equation.Note: For information onmultiplying and dividingnumbers in equations,see page 66.

3

4

5

6

CONTINUED

5 | 4 x + 2 | = 50

=

| 4 x + 2 | = 10

505

5 | 4 x + 2 |5

| 4 x + 2 | = 10

4 x + 2 = 10 4 x + 2 = –10



Solve an Equation with Absolute Values continued

74

Step 3: Solve the New Equations

4 x + 2 = 10

4 x + 2 – 2 = 10 – 2

4 x = 8

4 x + 2 = –1 0

4 x + 2 – 2 = –10 – 2

4 x = –12

You now need to solvethe two new equationsseparate ly in order todetermine the answerfor the original equation.In both equations,determine whichnumbers and variablesyou need to add orsubtract to place thevariable by itself on one

side of the equation.

Add or subtract thenumbers and/or variablesyou determined in step 7

on both sides of theequation.In both examples,subtract 2 from bothsides of the equation.

In both equations,determine whichnumbers you need tomultiply or divide byto place the variableby itself on one sideof the equation.

Multiply or divideby the numbers youdetermined in step 9

on both sides of theequation.In both examples,divide by 4 on bothsides of the equation.

•

• •

9 108

7

After you have created two new equationsfrom the original absolute value equation,you can solve each of the two new equationsand determine the two possible answers forthe 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 thevariable. When you place a number you foundfor the variable into the original equation and

work out the equation, both sides of theequation must equal each other in order tohave a solution.

Sometimes both of the numbers you foundfor the variable will solve the equation. Inother instances, one or both of your numberswill fail to solve the equation. This means that

either you have made a mistake and shoulddouble-check your calculations or there is onlyone or no solution for the equation.

4 x = –1 2

=

x = –3

–1 24

4 x

4

4 x = 8

=

x = 2

84

4 x

4




75

Step 4: Check Your Answers

P r a c t i c e Tip

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

1) | x | = 52) | 2x – 3 | = 53) 2 | x – 1 | = 8

4) 2 | 3x – 4 | – 3 = 55) | 3x – 9 | = 06) 3 | 2x + 6 | + 5 = 2

Is there a solution to every equation with absolute values?

No. As you work with equations thatcontain absolute values, you willsometimes come across equations thathave no solution. One dead give-awayis when the absolute value equals anegative number, as in | 4 x + 2 | = –10.Don’t even bother working out theseequations because no matter what x

equals, the absolute value of theexpression within the absolute valuesymbols cannot equal a negativenumber.

To check your answers, placethe first number you foundinto the original equationand solve the problem. Ifboth sides of the equationare equal, this numbercorrectly solves the equation.

11 Place the second number youfound into the original equationand solve the problem. If bothsides of the equation are equal,this number also correctly solvesthe equation.

12

If x = 2

5 | 4x + 2 | + 6 = 56

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

5 | 8 + 2 | + 6 = 56

5 | 10 | + 6 = 56

5 x 10 + 6 = 5656 = 56 Correct!

If x = –3

5 | 4x + 2 | + 6 = 56

5 | 4(–3) + 2 | + 6 = 56

5 | –12 + 2 | + 6 = 56

5 |–10 | + 6 = 56

5 x 10 + 6 = 5656 = 56 Correct!



Solve the following equations.Question 1.

Solving Basic Equations

a) 2 + x = 5

b) x – 4 = 10

c) x + 3 = 0

d) 2x = –8

e) 3x = 5

f) –x = –2

g) –8 x = –5

h)

i)

j) 2x = 1

k)


a) 2x + x + 1 = 2

b) 2x + 1 = x

c) 2x + 3 = 4 x + 4

d) 20 – x (4 + 1) = 5 x

e) 8x – 2(5 x + 6) = 20

f) 5x + 4(2 x – 7) = –15

Solve for y in the following equations.Question 3.

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)

Test Your Skills

= 105x3

= 33x7

= 6x4

76

S l i B i E tiCh t 3




a) | x | = 3

b) | – x | = 2

c) | x | = 0

d) | x | = –4

e) | x – 2 | = 1

f) | x – 10 | = 3

g) | x + 5 | = 5


a) | 2 x | = 4

b) | 3 x | = 8

c) | 2 x – 1 | = –1

d) | 3 x – 6 | = 0

e) | 2 x – 1 | = 1

f) | 5 x – 2 | + 4 = 6

g) 2 | 3 x – 1 | = 20

h) 4 | 2 x + 3 | + 4 = 20


You can check your answerson pages 268-269.

77

3 - Solving Basic Equations

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