Basic Mathematics for College Students

668 Chapter 8 An Introduction to Algebra

SECTION 8.4

More about Solving Equations

Objectives

1 Use more than one property of

equality to solve equations.

2 Simplify expressions to solve

equations.

We have solved simple equations by using properties of equality. We will now expand

our equation-solving skills by considering more complicated equations.

Use more than one property of equality to solve equations.

Sometimes we must use several properties of equality to solve an equation. For

example, on the left side of , the variable is multiplied by 2, and then 6

is added to that product. To isolate , we use the order of operations rules in reverse.

First, we undo the addition of 6, and then we undo the multiplication by 2.

This is the equation to solve.

To undo the addition of 6, subtract 6 from both sides.

Do the subtractions.

To undo the multiplication by 2, divide both sides by 2.

On the left side, simplify by removing the common

factor of 2 from the numerator and denominator:

x. On the right side, do the division.

The solution is 2.

2 x

1

(^21)
x 2
2 x
2



4

2

2 x 4

2 x 6  6  10  6

2 x 6  10

x

2 x 6  10 x

1

Self Check 1

Solve:

Now TryProblem 15

8 x 13  43

EXAMPLE (^1) Solve: 12 x 5  17
StrategyFirst we will use a property of equality to isolate the variable termon one
side of the equation. Then we will use a second property of equality to isolate the
variableitself.
WHYTo solve the original equation, we want to find a simpler equivalent
equation of the form , whose solution is obvious.
Solution
On the left side of the equation, is multiplied by 12, and then 5 is added to that
product. To isolate , we undo the operations in the opposite order.

• To isolate the variable term, 12 x,we subtract 5 from both sides to undo
  the addition of 5.


• To isolate the variable, , we divide both sides by 12 to undo the
  multiplication by 12.
  This is the equation to solve. First, we want to
  isolate the variable term, 12x.

On the left side simplify:.

On the right side, do the division.

12

1

x

x 1 121 x

Use the division property of equality: Divide both

(^) sides by 12 to isolate x.
12 x
12



12

12

Do the subtractions: 5 5 0 and 17 5 12.

12 x (^12) Now we want to isolate the variable, x.
Use the subtraction property of equality: Subtract 5
12 x 5  5  17  (^5) from both sides to isolate 12x.
12 x 5  17
x
x
x
xa number