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

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

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

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33

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

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.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44

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

Solutions of Equations

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.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55

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

Solutions of Equations

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.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66

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

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.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77

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

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.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88

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

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.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99

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

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.

?

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1010

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

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

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1111

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

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

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1212

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

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

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1313

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

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

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1414

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

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.

?