Algebra Examples

$\frac{2}{9} y = \frac{4}{3} y + 2 - \frac{4}{9} y$

Step 1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{4}{3} y + 2 - \frac{4}{9} y = \frac{2}{9} y$

Step 2

Simplify $\frac{4}{3} y + 2 - \frac{4}{9} y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{4}{3}$ and $y$ .

$\frac{4 y}{3} + 2 - \frac{4}{9} y = \frac{2}{9} y$

Step 2.1.2

Combine $y$ and $\frac{4}{9}$ .

$\frac{4 y}{3} + 2 - \frac{y \cdot 4}{9} = \frac{2}{9} y$

Step 2.1.3

Move $4$ to the left of $y$ .

$\frac{4 y}{3} + 2 - \frac{4 y}{9} = \frac{2}{9} y$

Step 2.2

To write $\frac{4 y}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 y}{3} \cdot \frac{3}{3} - \frac{4 y}{9} + 2 = \frac{2}{9} y$

Step 2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{4 y}{3}$ by $\frac{3}{3}$ .

$\frac{4 y \cdot 3}{3 \cdot 3} - \frac{4 y}{9} + 2 = \frac{2}{9} y$

Step 2.3.2

Multiply $3$ by $3$ .

$\frac{4 y \cdot 3}{9} - \frac{4 y}{9} + 2 = \frac{2}{9} y$

Step 2.4

Combine the numerators over the common denominator.

$\frac{4 y \cdot 3 - 4 y}{9} + 2 = \frac{2}{9} y$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $4 y$ out of $4 y \cdot 3 - 4 y$ .

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Step 2.5.1.1

Factor $4 y$ out of $4 y \cdot 3$ .

$\frac{4 y (3) - 4 y}{9} + 2 = \frac{2}{9} y$

Step 2.5.1.2

Factor $4 y$ out of $- 4 y$ .

$\frac{4 y (3) + 4 y (- 1)}{9} + 2 = \frac{2}{9} y$

Step 2.5.1.3

Factor $4 y$ out of $4 y (3) + 4 y (- 1)$ .

$\frac{4 y (3 - 1)}{9} + 2 = \frac{2}{9} y$

Step 2.5.2

Subtract $1$ from $3$ .

$\frac{4 y \cdot 2}{9} + 2 = \frac{2}{9} y$

Step 2.5.3

Multiply $2$ by $4$ .

$\frac{8 y}{9} + 2 = \frac{2}{9} y$

Step 3

Combine $\frac{2}{9}$ and $y$ .

$\frac{8 y}{9} + 2 = \frac{2 y}{9}$

Step 4

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

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Step 4.1

Subtract $\frac{2 y}{9}$ from both sides of the equation.

$\frac{8 y}{9} + 2 - \frac{2 y}{9} = 0$

Step 4.2

Combine the numerators over the common denominator.

$2 + \frac{8 y - 2 y}{9} = 0$

Step 4.3

Subtract $2 y$ from $8 y$ .

$2 + \frac{6 y}{9} = 0$

Step 4.4

Cancel the common factor of $6$ and $9$ .

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Step 4.4.1

Factor $3$ out of $6 y$ .

$2 + \frac{3 (2 y)}{9} = 0$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $3$ out of $9$ .

$2 + \frac{3 (2 y)}{3 (3)} = 0$

Step 4.4.2.2

Cancel the common factor.

$2 + \frac{3 (2 y)}{3 \cdot 3} = 0$

Step 4.4.2.3

Rewrite the expression.

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

Step 5

Subtract $2$ from both sides of the equation.

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

Step 6

Multiply both sides of the equation by $\frac{3}{2}$ .

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

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{3}{2} \cdot \frac{2 y}{3}$ .

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

Cancel the common factor.

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

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{3}{2} \cdot - 2$

Step 7.1.1.2

Cancel the common factor of $2$ .

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Step 7.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{3}{2} \cdot - 2$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{3}{2} \cdot - 2$

Step 7.1.1.2.3

Rewrite the expression.

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

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{3}{2} \cdot - 2$ .

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Factor $2$ out of $- 2$ .

$y = \frac{3}{2} \cdot (2 (- 1))$

Step 7.2.1.1.2

Cancel the common factor.

$y = \frac{3}{2} \cdot (2 \cdot - 1)$

Step 7.2.1.1.3

Rewrite the expression.

$y = 3 \cdot - 1$

Step 7.2.1.2

Multiply $3$ by $- 1$ .

$y = - 3$