Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.6 Solving Equations: The Addition and Multiplication Properties.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

2.6

Solving Equations: The Addition and

Multiplication Properties

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22


Statements like 5 + 2 = 7 are called equations.

An equation is of the form expression = expression.

An equation can be labeled as

Equal sign

left side right side

x + 5 = 9

Equations



Solutions of Equations

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

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




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.




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.



Solving Equations

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

x = number or number = x

Equivalent equations have the same solution.

The word “number” above represents the solution of the original equation.



Addition Property of Equality

Let a, b, and c represent numbers.

If a = b, then

a + c = b + c

and

a – c = b c

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



Solve for x.

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

equation in the form x = number. To do so, we add 4 to both sides of the equation. x 4 = 3 x 4 + 4 = 3 + 4 Add 4 to both sides. x = 7 Simplify.



Check

x 4 = 3 Original equation

7 4 = 3 Replace x with 7.

3 = 3 True.

Since 3 = 3 is a true statement, 7 is the solution of the equation.

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

?



Note that it is always a good idea to check the solution in the original equation to see that it makes the equation a true statement.

Helpful Hint



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

x = 3 and 3 = xboth have a solution of 3.

Helpful Hint



Multiplication Property of Equality

Let a, b, and c represent numbers and let c ≠ 0. If a = b, then

a • c = b • c and

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

a b=

c c



Solve for x

4x = 8To solve the equation for x, notice that 4 is multiplied by x.

To get x alone, we divide both sides of the equation by 4 and then simplify.

4 8

4 4

x= 11∙∙xx = 2 or = 2 or xx = 2 = 2



Check

To check, replace x with 2 in the original equation.

4x = 8 Original equation

4 • 2 = 8 Let x = 2.

8 = 8 True.

The solution is 2.

?

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.6 Solving Equations: The Addition and Multiplication Properties.

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