Finite Math Examples

$\frac{2}{3} y + \frac{3}{4} y - 5 = \frac{2}{3} y + 4$

Step 1

Simplify $\frac{2}{3} y + \frac{3}{4} y - 5$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{2 y}{3} + \frac{3}{4} y - 5 = \frac{2}{3} y + 4$

Step 1.1.2

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

$\frac{2 y}{3} + \frac{3 y}{4} - 5 = \frac{2}{3} y + 4$

Step 1.2

To write $\frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 y}{3} \cdot \frac{4}{4} + \frac{3 y}{4} - 5 = \frac{2}{3} y + 4$

Step 1.3

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

$\frac{2 y}{3} \cdot \frac{4}{4} + \frac{3 y}{4} \cdot \frac{3}{3} - 5 = \frac{2}{3} y + 4$

Step 1.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 y}{3}$ by $\frac{4}{4}$ .

$\frac{2 y \cdot 4}{3 \cdot 4} + \frac{3 y}{4} \cdot \frac{3}{3} - 5 = \frac{2}{3} y + 4$

Step 1.4.2

Multiply $3$ by $4$ .

$\frac{2 y \cdot 4}{12} + \frac{3 y}{4} \cdot \frac{3}{3} - 5 = \frac{2}{3} y + 4$

Step 1.4.3

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

$\frac{2 y \cdot 4}{12} + \frac{3 y \cdot 3}{4 \cdot 3} - 5 = \frac{2}{3} y + 4$

Step 1.4.4

Multiply $4$ by $3$ .

$\frac{2 y \cdot 4}{12} + \frac{3 y \cdot 3}{12} - 5 = \frac{2}{3} y + 4$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 y \cdot 4 + 3 y \cdot 3}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $y$ out of $2 y \cdot 4 + 3 y \cdot 3$ .

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Step 1.6.1.1.1

Factor $y$ out of $2 y \cdot 4$ .

$\frac{y (2 \cdot 4) + 3 y \cdot 3}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6.1.1.2

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

$\frac{y (2 \cdot 4) + y (3 \cdot 3)}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6.1.1.3

Factor $y$ out of $y (2 \cdot 4) + y (3 \cdot 3)$ .

$\frac{y (2 \cdot 4 + 3 \cdot 3)}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6.1.2

Multiply $2$ by $4$ .

$\frac{y (8 + 3 \cdot 3)}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6.1.3

Multiply $3$ by $3$ .

$\frac{y (8 + 9)}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6.1.4

Add $8$ and $9$ .

$\frac{y \cdot 17}{12} - 5 = \frac{2}{3} y + 4$

Step 1.6.2

Move $17$ to the left of $y$ .

$\frac{17 y}{12} - 5 = \frac{2}{3} y + 4$

Step 2

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

$\frac{17 y}{12} - 5 = \frac{2 y}{3} + 4$

Step 3

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

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Step 3.1

Subtract $\frac{2 y}{3}$ from both sides of the equation.

$\frac{17 y}{12} - 5 - \frac{2 y}{3} = 4$

Step 3.2

To write $- \frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{17 y}{12} - \frac{2 y}{3} \cdot \frac{4}{4} - 5 = 4$

Step 3.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{2 y}{3}$ by $\frac{4}{4}$ .

$\frac{17 y}{12} - \frac{2 y \cdot 4}{3 \cdot 4} - 5 = 4$

Step 3.3.2

Multiply $3$ by $4$ .

$\frac{17 y}{12} - \frac{2 y \cdot 4}{12} - 5 = 4$

Step 3.4

Combine the numerators over the common denominator.

$\frac{17 y - 2 y \cdot 4}{12} - 5 = 4$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $y$ out of $17 y - 2 y \cdot 4$ .

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Step 3.5.1.1.1

Factor $y$ out of $17 y$ .

$\frac{y \cdot 17 - 2 y \cdot 4}{12} - 5 = 4$

Step 3.5.1.1.2

Factor $y$ out of $- 2 y \cdot 4$ .

$\frac{y \cdot 17 + y (- 2 \cdot 4)}{12} - 5 = 4$

Step 3.5.1.1.3

Factor $y$ out of $y \cdot 17 + y (- 2 \cdot 4)$ .

$\frac{y (17 - 2 \cdot 4)}{12} - 5 = 4$

Step 3.5.1.2

Multiply $- 2$ by $4$ .

$\frac{y (17 - 8)}{12} - 5 = 4$

Step 3.5.1.3

Subtract $8$ from $17$ .

$\frac{y \cdot 9}{12} - 5 = 4$

Step 3.5.2

Cancel the common factor of $9$ and $12$ .

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Step 3.5.2.1

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

$\frac{3 (y \cdot 3)}{12} - 5 = 4$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $3$ out of $12$ .

$\frac{3 (y \cdot 3)}{3 (4)} - 5 = 4$

Step 3.5.2.2.2

Cancel the common factor.

$\frac{3 (y \cdot 3)}{3 \cdot 4} - 5 = 4$

Step 3.5.2.2.3

Rewrite the expression.

$\frac{y \cdot 3}{4} - 5 = 4$

Step 3.5.3

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

$\frac{3 y}{4} - 5 = 4$

Step 4

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

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Step 4.1

Add $5$ to both sides of the equation.

$\frac{3 y}{4} = 4 + 5$

Step 4.2

Add $4$ and $5$ .

$\frac{3 y}{4} = 9$

Step 5

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} \cdot 9$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{4}{3} \cdot \frac{3 y}{4}$ .

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Step 6.1.1.1

Cancel the common factor of $4$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} \cdot 9$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{4}{3} \cdot 9$

Step 6.1.1.2

Cancel the common factor of $3$ .

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Step 6.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{4}{3} \cdot 9$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{4}{3} \cdot 9$

Step 6.1.1.2.3

Rewrite the expression.

$y = \frac{4}{3} \cdot 9$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{4}{3} \cdot 9$ .

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Factor $3$ out of $9$ .

$y = \frac{4}{3} \cdot (3 (3))$

Step 6.2.1.1.2

Cancel the common factor.

$y = \frac{4}{3} \cdot (3 \cdot 3)$

Step 6.2.1.1.3

Rewrite the expression.

$y = 4 \cdot 3$

Step 6.2.1.2

Multiply $4$ by $3$ .

$y = 12$