Solving Equations: The Addition Solving Equations: The Addition and Multiplication Propertiesand Multiplication Properties

Section 2.6

EquationEquation

Statements like 5 Statements like 5 ++ 2 2 == 7 are called 7 are called equationsequations..

An equation is of the form An equation is of the form expression expression == expression expression

An equation can be labeled as An equation can be labeled as Equal sign

left side right side

x + 5 = 9

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Solving/SolutionSolving/Solution

When an equation contains a variable, When an equation contains a variable, deciding which values of the variable deciding which values of the variable make an equation a true statement is make an equation a true statement is called called solvingsolving an equation for the an equation for the variable.variable.

A A solutionsolution of an equation is a value for of an equation is a value for the variable that makes an equation a the variable that makes an equation a true statement.true statement.

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Solving/Solution ...Solving/Solution ...

Determine whether a number is a solution:Determine whether a number is a solution:

Is -2 a solution of the equation 2y + 1 = -3?

Replace y with -2 in the equation.

2y + 1 = -3

2(-2) + 1 = -3?

- 4 + 1 = -3

-3 = -3

?

TrueTrue

Since -3 = -3 is a true statement, -2 is a solution of the equation.

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Solving/Solution ...Solving/Solution ...

Determine whether a number is a solution:Determine whether a number is a solution:

Is 6 a solution of the equation 5x - 1 = 30?

Replace x with 6 in the equation.

5x - 1 = 30

5(6) - 1 = 30?

30 - 1 = 30

29 = 30

?

FalseFalse

Since 29 = 30 is a false statement, 6 is not a solution of the equation.

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To solve an equation, we will use To solve an equation, we will use properties of equality to write simpler properties of equality to write simpler equations, all equivalent to the original equations, all equivalent to the original equation, until the final equation has the equation, until the final equation has the form form xx == number number or or number number == xx

Equivalent equationsEquivalent equations have the have the samesame solutionsolution. . The word “number” above represents the The word “number” above represents the solution of the original equation.solution of the original equation.

Solving/Solution...Solving/Solution...

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Addition Property of EqualityAddition Property of Equality

Let Let aa, , bb, and c represent numbers., and c represent numbers.If If a a == bb, then, then

aa ++ cc == bb ++ c c andand

a a –– c c == b b c c

In other words, the same number may be In other words, the same number may be added to or subtracted from both sides added to or subtracted from both sides of an equation without changing the of an equation without changing the solution of the equation.solution of the equation.

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Solve for Solve for xx..

xx 4 4 == 3 3To solve the equation for To solve the equation for xx, we need to rewrite , we need to rewrite

the equation in the form the equation in the form xx == number. number. To do so, we add 4 to both sides of the To do so, we add 4 to both sides of the

equation.equation. xx 4 4 == 3 3 xx 4 4 ++ 4 4 == 3 3 ++ 4 4 Add Add 44 to both sides. to both sides. xx == 7 Simplify. 7 Simplify.

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Check Check

xx 4 4 == 3 Original equation 3 Original equation

77 4 4 == 3 Replace 3 Replace xx with with 77..

3 3 == 3 3 True. True.

Since 3 Since 3 == 3 is a true statement, 3 is a true statement, 77 is the is the solutionsolution of the equation. of the equation.

To check, replace x with 7 in the original equation.

?

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Remember to check the solution in the original equation to see that it makes the equation a true statement.

Helpful HintHelpful Hint

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Remember that we can get the variable alone on either side of the equation. For example, the equations

x = 3 and 3 = x

both have a solution of 3.

Helpful HintHelpful Hint

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Multiplication Property of EqualityMultiplication Property of Equality

Let Let aa, , bb, and , and cc represent numbers and let represent numbers and let cc 0. If 0. If aa == bb, then, then

a c = b c and and

In other words, both sides of an equation may be multiplied In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing or divided by the same nonzero number without changing the solution of the equation.the solution of the equation.

a bc c

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Solve for Solve for xx

44xx == 8 8To solve the equation for To solve the equation for xx, notice that 4 is , notice that 4 is

multipliedmultiplied by by xx..

To get To get xx alone, we alone, we dividedivide both sides of the both sides of the equation by equation by 44 and then simplify. and then simplify.

4

4 4

8

x

11xx = 2 or = 2 or xx = 2 = 2

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Check Check

To To checkcheck, replace , replace xx with 2 in the with 2 in the original equationoriginal equation..

44xx == 8 Original equation 8 Original equation

4 4 22 == 8 Let 8 Let xx == 22..

8 8 == 8 True. 8 True.

The The solutionsolution is is 22..

?

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As reviewed in Chapter 1, don’t forget that order is important when subtracting. Notice the translation order of numbers and variables below.

Helpful HintHelpful Hint

PhrasePhrase

a number less 9a number subtracted from 9

TranslationTranslation

x - 99 - x

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