Finite Math Examples

$\frac{5 y}{6} - 8 = 5 - \frac{2 y}{3}$

Step 1

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

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Step 1.1

Add $\frac{2 y}{3}$ to both sides of the equation.

$\frac{5 y}{6} - 8 + \frac{2 y}{3} = 5$

Step 1.2

To write $\frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 y}{6} + \frac{2 y}{3} \cdot \frac{2}{2} - 8 = 5$

Step 1.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 y}{3}$ by $\frac{2}{2}$ .

$\frac{5 y}{6} + \frac{2 y \cdot 2}{3 \cdot 2} - 8 = 5$

Step 1.3.2

Multiply $3$ by $2$ .

$\frac{5 y}{6} + \frac{2 y \cdot 2}{6} - 8 = 5$

Step 1.4

Combine the numerators over the common denominator.

$\frac{5 y + 2 y \cdot 2}{6} - 8 = 5$

Step 1.5

Simplify each term.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Factor $y$ out of $5 y + 2 y \cdot 2$ .

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Step 1.5.1.1.1

Factor $y$ out of $5 y$ .

$\frac{y \cdot 5 + 2 y \cdot 2}{6} - 8 = 5$

Step 1.5.1.1.2

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

$\frac{y \cdot 5 + y (2 \cdot 2)}{6} - 8 = 5$

Step 1.5.1.1.3

Factor $y$ out of $y \cdot 5 + y (2 \cdot 2)$ .

$\frac{y (5 + 2 \cdot 2)}{6} - 8 = 5$

Step 1.5.1.2

Multiply $2$ by $2$ .

$\frac{y (5 + 4)}{6} - 8 = 5$

Step 1.5.1.3

Add $5$ and $4$ .

$\frac{y \cdot 9}{6} - 8 = 5$

Step 1.5.2

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

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Step 1.5.2.1

Factor $3$ out of $y \cdot 9$ .

$\frac{3 (y \cdot 3)}{6} - 8 = 5$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $3$ out of $6$ .

$\frac{3 (y \cdot 3)}{3 (2)} - 8 = 5$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{3 (y \cdot 3)}{3 \cdot 2} - 8 = 5$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{y \cdot 3}{2} - 8 = 5$

Step 1.5.3

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

$\frac{3 y}{2} - 8 = 5$

Step 2

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

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Step 2.1

Add $8$ to both sides of the equation.

$\frac{3 y}{2} = 5 + 8$

Step 2.2

Add $5$ and $8$ .

$\frac{3 y}{2} = 13$

Step 3

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

$\frac{2}{3} \cdot \frac{3 y}{2} = \frac{2}{3} \cdot 13$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 y}{2} = \frac{2}{3} \cdot 13$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{2}{3} \cdot 13$

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{2}{3} \cdot 13$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{2}{3} \cdot 13$

Step 4.1.1.2.3

Rewrite the expression.

$y = \frac{2}{3} \cdot 13$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $\frac{2}{3} \cdot 13$ .

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Step 4.2.1.1

Combine $\frac{2}{3}$ and $13$ .

$y = \frac{2 \cdot 13}{3}$

Step 4.2.1.2

Multiply $2$ by $13$ .

$y = \frac{26}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{26}{3}$

Decimal Form:

$y = 8.\overline{6}$

Mixed Number Form:

$y = 8 \frac{2}{3}$