Precalculus Examples

$| y - 31 | - | y - 39 | = 0$

Step 1

Subtract $| y - 31 |$ from both sides of the equation.

$- | y - 39 | = - | y - 31 |$

Step 2

Divide each term in $- | y - 39 | = - | y - 31 |$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- | y - 39 | = - | y - 31 |$ by $- 1$ .

$\frac{- | y - 39 |}{- 1} = \frac{- | y - 31 |}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{| y - 39 |}{1} = \frac{- | y - 31 |}{- 1}$

Step 2.2.2

Divide $| y - 39 |$ by $1$ .

$| y - 39 | = \frac{- | y - 31 |}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

$| y - 39 | = \frac{| y - 31 |}{1}$

Step 2.3.2

Divide $| y - 31 |$ by $1$ .

$| y - 39 | = | y - 31 |$

Step 3

Rewrite the absolute value equation as four equations without absolute value bars.

$y - 39 = y - 31$

$y - 39 = - (y - 31)$

$- (y - 39) = y - 31$

$- (y - 39) = - (y - 31)$

Step 4

After simplifying, there are only two unique equations to be solved.

$y - 39 = y - 31$

$y - 39 = - (y - 31)$

Step 5

Solve $y - 39 = y - 31$ for $y$ .

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Step 5.1

Move all terms containing $y$ to the left side of the equation.

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Step 5.1.1

Subtract $y$ from both sides of the equation.

$y - 39 - y = - 31$

Step 5.1.2

Combine the opposite terms in $y - 39 - y$ .

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Step 5.1.2.1

Subtract $y$ from $y$ .

$0 - 39 = - 31$

Step 5.1.2.2

Subtract $39$ from $0$ .

$- 39 = - 31$

Step 5.2

Since $- 39 \neq - 31$ , there are no solutions.

No solution

Step 6

Solve $y - 39 = - (y - 31)$ for $y$ .

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Step 6.1

Simplify $- (y - 31)$ .

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Step 6.1.1

Rewrite.

$y - 39 = 0 + 0 - (y - 31)$

Step 6.1.2

Simplify by adding zeros.

$y - 39 = - (y - 31)$

Step 6.1.3

Apply the distributive property.

$y - 39 = - y - - 31$

Step 6.1.4

Multiply $- 1$ by $- 31$ .

$y - 39 = - y + 31$

Step 6.2

Move all terms containing $y$ to the left side of the equation.

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Step 6.2.1

Add $y$ to both sides of the equation.

$y - 39 + y = 31$

Step 6.2.2

Add $y$ and $y$ .

$2 y - 39 = 31$

Step 6.3

Move all terms not containing $y$ to the right side of the equation.

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Step 6.3.1

Add $39$ to both sides of the equation.

$2 y = 31 + 39$

Step 6.3.2

Add $31$ and $39$ .

$2 y = 70$

Step 6.4

Divide each term in $2 y = 70$ by $2$ and simplify.

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Step 6.4.1

Divide each term in $2 y = 70$ by $2$ .

$\frac{2 y}{2} = \frac{70}{2}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $2$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{70}{2}$

Step 6.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{70}{2}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Divide $70$ by $2$ .

$y = 35$

Step 7