Basic Mathematics for College Students

Use the subtraction property of equality.

To introduce another property of equality, consider
the first scale shown on the right, which represents
the equation x 3 5. The scale is in balance
because the weights on the left and right sides are
equal.To find x, we need to remove 3 from the left
side. To keep the scale in balance, we must also
remove 3 from the right side. After doing this, we
see that xis balanced by 2. Therefore,xmust be 2.
We say that we have solved the equation x  3  5
and that the solution is 2. This example illustrates
the following property of equality.

Subtraction Property of Equality

Subtracting the same number from both sides of an equation does not

change its solution.

For any numbers , , and ,

if , then

When we use this property of equality, the resulting equation is equivalent to the
original one.

ab acbc

ab c

x + 3 = 5

Remove

3

Remove

3

x = 2

11

111 11111

x

x

3

8.3 Solving Equations Using Properties of Equality 661

Caution! After checking a result, be careful when stating your conclusion.

Here, it would be incorrect to say:

The solution is.

The number we were checking was 24, not  3.

 3

Self Check 4

Solve:

a.

b.

Now TryProblems 49 and 51

0.7a0.2

x

4

15



11

5

EXAMPLE 4

Solve: a. b.

StrategyWe will use a property of equality to isolate the variable on one side of
the equation.

WHYTo solve the original equation, we want to find a simpler equivalent
equation of the form , whose solution is obvious.

Solution
a.To isolate , we use the subtraction property of equality. We can undo the
addition of by subtracting from both sides.

This is the equation to solve.

Subtract from both sides.

On the left side,.

On the right side, build so that it has a

denominator of 8.

Multiply the numerators and multiply the denominators.

Subtract the numerators. Write the result over the

common denominator 8.

The solution is. Check by substituting it for in the original equation.x

13

8

x

13

8

x

14

8



1

8

x^7 47

4



2

2



1

8

x  81  81  0

7

4



1

8

x 81

1

8



1

8



7

4



1

8

x

1

8



7

4

18 18

x

xa number

x 54.9x45.2

1

8



7

4