Precalculus Examples

$| 7 y - 6 | = | 6 - 7 y |$

Step 1

Rewrite the equation as $| 6 - 7 y | = | 7 y - 6 |$ .

$| 6 - 7 y | = | 7 y - 6 |$

Step 2

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

$6 - 7 y = 7 y - 6$

$6 - 7 y = - (7 y - 6)$

$- (6 - 7 y) = 7 y - 6$

$- (6 - 7 y) = - (7 y - 6)$

Step 3

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

$6 - 7 y = 7 y - 6$

$6 - 7 y = - (7 y - 6)$

Step 4

Solve $6 - 7 y = 7 y - 6$ for $y$ .

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Step 4.1

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

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Step 4.1.1

Subtract $7 y$ from both sides of the equation.

$6 - 7 y - 7 y = - 6$

Step 4.1.2

Subtract $7 y$ from $- 7 y$ .

$6 - 14 y = - 6$

Step 4.2

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

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Step 4.2.1

Subtract $6$ from both sides of the equation.

$- 14 y = - 6 - 6$

Step 4.2.2

Subtract $6$ from $- 6$ .

$- 14 y = - 12$

Step 4.3

Divide each term in $- 14 y = - 12$ by $- 14$ and simplify.

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Step 4.3.1

Divide each term in $- 14 y = - 12$ by $- 14$ .

$\frac{- 14 y}{- 14} = \frac{- 12}{- 14}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 14$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 14 y}{- 14} = \frac{- 12}{- 14}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 12}{- 14}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $- 12$ and $- 14$ .

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Step 4.3.3.1.1

Factor $- 2$ out of $- 12$ .

$y = \frac{- 2 (6)}{- 14}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $- 2$ out of $- 14$ .

$y = \frac{- 2 \cdot 6}{- 2 \cdot 7}$

Step 4.3.3.1.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 6}{- 2 \cdot 7}$

Step 4.3.3.1.2.3

Rewrite the expression.

$y = \frac{6}{7}$

Step 5

Solve $6 - 7 y = - (7 y - 6)$ for $y$ .

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Step 5.1

Simplify $- (7 y - 6)$ .

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Step 5.1.1

Rewrite.

$6 - 7 y = 0 + 0 - (7 y - 6)$

Step 5.1.2

Simplify by adding zeros.

$6 - 7 y = - (7 y - 6)$

Step 5.1.3

Apply the distributive property.

$6 - 7 y = - (7 y) - - 6$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $7$ by $- 1$ .

$6 - 7 y = - 7 y - - 6$

Step 5.1.4.2

Multiply $- 1$ by $- 6$ .

$6 - 7 y = - 7 y + 6$

Step 5.2

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

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Step 5.2.1

Add $7 y$ to both sides of the equation.

$6 - 7 y + 7 y = 6$

Step 5.2.2

Combine the opposite terms in $6 - 7 y + 7 y$ .

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Step 5.2.2.1

Add $- 7 y$ and $7 y$ .

$6 + 0 = 6$

Step 5.2.2.2

Add $6$ and $0$ .

$6 = 6$

Step 5.3

Since $6 = 6$ , the equation will always be true.

All real numbers

Step 6

List all of the solutions.

$y = \frac{6}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{6}{7}$

Decimal Form:

$y = 0.\overline{857142}$