Pre-Algebra Examples

$| 2 y | = | \frac{1}{2} y |$

Step 1

Rewrite the equation as $| \frac{1}{2} y | = | 2 y |$ .

$| \frac{1}{2} y | = | 2 y |$

Step 2

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

$\frac{1}{2} y = 2 y$

$\frac{1}{2} y = - 1 \cdot (2 y)$

$- \frac{1}{2} y = 2 y$

$- \frac{1}{2} y = - 1 \cdot (2 y)$

Step 3

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

$\frac{1}{2} y = 2 y$

$\frac{1}{2} y = - 1 \cdot (2 y)$

Step 4

Solve $\frac{1}{2} y = 2 y$ for $y$ .

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Step 4.1

Simplify $\frac{1}{2} y$ .

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Step 4.1.1

Rewrite.

$0 + 0 + \frac{1}{2} y = 2 y$

Step 4.1.2

Simplify by adding zeros.

$\frac{1}{2} y = 2 y$

Step 4.1.3

Combine $\frac{1}{2}$ and $y$ .

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

Step 4.2

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

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Step 4.2.1

Subtract $2 y$ from both sides of the equation.

$\frac{y}{2} - 2 y = 0$

Step 4.2.2

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{2} - 2 y \cdot \frac{2}{2} = 0$

Step 4.2.3

Combine $- 2 y$ and $\frac{2}{2}$ .

$\frac{y}{2} + \frac{- 2 y \cdot 2}{2} = 0$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{y - 2 y \cdot 2}{2} = 0$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Factor $y$ out of $y - 2 y \cdot 2$ .

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Step 4.2.5.1.1

Raise $y$ to the power of $1$ .

$\frac{y^{1} - 2 y \cdot 2}{2} = 0$

Step 4.2.5.1.2

Factor $y$ out of $y^{1}$ .

$\frac{y \cdot 1 - 2 y \cdot 2}{2} = 0$

Step 4.2.5.1.3

Factor $y$ out of $- 2 y \cdot 2$ .

$\frac{y \cdot 1 + y (- 2 \cdot 2)}{2} = 0$

Step 4.2.5.1.4

Factor $y$ out of $y \cdot 1 + y (- 2 \cdot 2)$ .

$\frac{y (1 - 2 \cdot 2)}{2} = 0$

Step 4.2.5.2

Multiply $- 2$ by $2$ .

$\frac{y (1 - 4)}{2} = 0$

Step 4.2.5.3

Subtract $4$ from $1$ .

$\frac{y \cdot - 3}{2} = 0$

Step 4.2.6

Move $- 3$ to the left of $y$ .

$\frac{- 3 \cdot y}{2} = 0$

Step 4.2.7

Move the negative in front of the fraction.

$- \frac{3 y}{2} = 0$

Step 4.3

Set the numerator equal to zero.

$3 y = 0$

Step 4.4

Divide each term in $3 y = 0$ by $3$ and simplify.

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Step 4.4.1

Divide each term in $3 y = 0$ by $3$ .

$\frac{3 y}{3} = \frac{0}{3}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{0}{3}$

Step 4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{3}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $3$ .

$y = 0$

Step 5

List all of the solutions.

$y = 0, 0$