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Solve: 6 ( p 3 ) 7 = 5 ( 4 p + 3 ) 12 .

p = −2

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Solve: 8 ( q + 1 ) 5 = 3 ( 2 q 4 ) 1 .

q = −8

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Solve: 10 [ 3 8 ( 2 s 5 ) ] = 15 ( 40 5 s ) .

Solution

.
Simplify from the innermost parentheses first. .
Combine like terms in the brackets. .
Distribute. .
Add 160 s to get the s’s to the right. .
Simplify. .
Subtract 600 to get the constants to the left. .
Simplify. .
Divide. .
Simplify. .
Check: .
Substitute s = −2 . .
.
.
.
.
.
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Solve: 6 [ 4 2 ( 7 y 1 ) ] = 8 ( 13 8 y ) .

y = 17 5

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Solve: 12 [ 1 5 ( 4 z 1 ) ] = 3 ( 24 + 11 z ) .

z = 0

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Solve: 0.36 ( 100 n + 5 ) = 0.6 ( 30 n + 15 ) .

Solution

.
Distribute. .
Subtract 18 n to get the variables to the left. .
Simplify. .
Subtract 1.8 to get the constants to the right. .
Simplify. .
Divide. .
Simplify. .
Check: .
Let n = 0.4 . .
.
.
.
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Solve: 0.55 ( 100 n + 8 ) = 0.6 ( 85 n + 14 ) .

n = 1

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Solve: 0.15 ( 40 m 120 ) = 0.5 ( 60 m + 12 ) .

m = −1

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Classify equations

Consider the equation we solved at the start of the last section, 7 x + 8 = −13 . The solution we found was x = −3 . This means the equation 7 x + 8 = −13 is true when we replace the variable, x , with the value −3 . We showed this when we checked the solution x = −3 and evaluated 7 x + 8 = −13 for x = −3 .

This figure shows why we can say the equation 7x plus 8 equals negative 13 is true when the variable x is replaced with the value negative 3. The first line shows the equation with negative 3 substituted in for x: 7 times negative 3 plus 8 might equal negative 13. Below this is the equation negative 21 plus 8 might equal negative 13. Below this is the equation negative 13 equals negative 13, with a check mark next to it.

If we evaluate 7 x + 8 for a different value of x , the left side will not be −13 .

The equation 7 x + 8 = −13 is true when we replace the variable, x , with the value −3 , but not true when we replace x with any other value. Whether or not the equation 7 x + 8 = −13 is true depends on the value of the variable. Equations like this are called conditional equations.

All the equations we have solved so far are conditional equations.

Conditional equation

An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation    .

Now let’s consider the equation 2 y + 6 = 2 ( y + 3 ) . Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y .

.
Distribute. .
Subtract 2 y to get the y ’s to one side. .
Simplify—the y ’s are gone! .

But 6 = 6 is true.

This means that the equation 2 y + 6 = 2 ( y + 3 ) is true for any value of y . We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable like this is called an identity    .

Identity

An equation that is true for any value of the variable is called an identity    .

The solution of an identity is all real numbers.

What happens when we solve the equation 5 z = 5 z 1 ?

.
Subtract 5 z to get the constant alone on the right. .
Simplify—the z ’s are gone! .

But 0 1 .

Solving the equation 5 z = 5 z 1 led to the false statement 0 = −1 . The equation 5 z = 5 z 1 will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction    .

Contradiction

An equation that is false for all values of the variable is called a contradiction    .

A contradiction has no solution.

Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.

6 ( 2 n 1 ) + 3 = 2 n 8 + 5 ( 2 n + 1 )

Solution

.
Distribute. .
Combine like terms. .
Subtract 12 n to get the n ’s to one side. .
Simplify. .
This is a true statement. The equation is an identity.
The solution is all real numbers.
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Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:

4 + 9 ( 3 x 7 ) = −42 x 13 + 23 ( 3 x 2 )

identity; all real numbers

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Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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