Basic Math Examples

$| \frac{4}{5} y - 7 | = | \frac{3}{4} y + 5 |$

Step 1

Rewrite the equation as $| \frac{3}{4} y + 5 | = | \frac{4}{5} y - 7 |$ .

$| \frac{3}{4} y + 5 | = | \frac{4}{5} y - 7 |$

Step 2

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

$\frac{3}{4} y + 5 = \frac{4}{5} y - 7$

$\frac{3}{4} y + 5 = - (\frac{4}{5} y - 7)$

$- (\frac{3}{4} y + 5) = \frac{4}{5} y - 7$

$- (\frac{3}{4} y + 5) = - (\frac{4}{5} y - 7)$

Step 3

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

$\frac{3}{4} y + 5 = \frac{4}{5} y - 7$

$\frac{3}{4} y + 5 = - (\frac{4}{5} y - 7)$

Step 4

Solve $\frac{3}{4} y + 5 = \frac{4}{5} y - 7$ for $y$ .

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Step 4.1

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

$\frac{3 y}{4} + 5 = \frac{4}{5} y - 7$

Step 4.2

Combine $\frac{4}{5}$ and $y$ .

$\frac{3 y}{4} + 5 = \frac{4 y}{5} - 7$

Step 4.3

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

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Step 4.3.1

Subtract $\frac{4 y}{5}$ from both sides of the equation.

$\frac{3 y}{4} + 5 - \frac{4 y}{5} = - 7$

Step 4.3.2

To write $\frac{3 y}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 y}{4} \cdot \frac{5}{5} - \frac{4 y}{5} + 5 = - 7$

Step 4.3.3

To write $- \frac{4 y}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 y}{4} \cdot \frac{5}{5} - \frac{4 y}{5} \cdot \frac{4}{4} + 5 = - 7$

Step 4.3.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.4.1

Multiply $\frac{3 y}{4}$ by $\frac{5}{5}$ .

$\frac{3 y \cdot 5}{4 \cdot 5} - \frac{4 y}{5} \cdot \frac{4}{4} + 5 = - 7$

Step 4.3.4.2

Multiply $4$ by $5$ .

$\frac{3 y \cdot 5}{20} - \frac{4 y}{5} \cdot \frac{4}{4} + 5 = - 7$

Step 4.3.4.3

Multiply $\frac{4 y}{5}$ by $\frac{4}{4}$ .

$\frac{3 y \cdot 5}{20} - \frac{4 y \cdot 4}{5 \cdot 4} + 5 = - 7$

Step 4.3.4.4

Multiply $5$ by $4$ .

$\frac{3 y \cdot 5}{20} - \frac{4 y \cdot 4}{20} + 5 = - 7$

Step 4.3.5

Combine the numerators over the common denominator.

$\frac{3 y \cdot 5 - 4 y \cdot 4}{20} + 5 = - 7$

Step 4.3.6

Simplify each term.

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Step 4.3.6.1

Simplify the numerator.

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Step 4.3.6.1.1

Factor $y$ out of $3 y \cdot 5 - 4 y \cdot 4$ .

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Step 4.3.6.1.1.1

Factor $y$ out of $3 y \cdot 5$ .

$\frac{y (3 \cdot 5) - 4 y \cdot 4}{20} + 5 = - 7$

Step 4.3.6.1.1.2

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

$\frac{y (3 \cdot 5) + y (- 4 \cdot 4)}{20} + 5 = - 7$

Step 4.3.6.1.1.3

Factor $y$ out of $y (3 \cdot 5) + y (- 4 \cdot 4)$ .

$\frac{y (3 \cdot 5 - 4 \cdot 4)}{20} + 5 = - 7$

Step 4.3.6.1.2

Multiply $3$ by $5$ .

$\frac{y (15 - 4 \cdot 4)}{20} + 5 = - 7$

Step 4.3.6.1.3

Multiply $- 4$ by $4$ .

$\frac{y (15 - 16)}{20} + 5 = - 7$

Step 4.3.6.1.4

Subtract $16$ from $15$ .

$\frac{y \cdot - 1}{20} + 5 = - 7$

Step 4.3.6.2

Move $- 1$ to the left of $y$ .

$\frac{- 1 \cdot y}{20} + 5 = - 7$

Step 4.3.6.3

Move the negative in front of the fraction.

$- \frac{y}{20} + 5 = - 7$

Step 4.4

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

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Step 4.4.1

Subtract $5$ from both sides of the equation.

$- \frac{y}{20} = - 7 - 5$

Step 4.4.2

Subtract $5$ from $- 7$ .

$- \frac{y}{20} = - 12$

Step 4.5

Multiply both sides of the equation by $- 20$ .

$- 20 (- \frac{y}{20}) = - 20 \cdot - 12$

Step 4.6

Simplify both sides of the equation.

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Step 4.6.1

Simplify the left side.

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Step 4.6.1.1

Simplify $- 20 (- \frac{y}{20})$ .

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Step 4.6.1.1.1

Cancel the common factor of $20$ .

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Step 4.6.1.1.1.1

Move the leading negative in $- \frac{y}{20}$ into the numerator.

$- 20 \frac{- y}{20} = - 20 \cdot - 12$

Step 4.6.1.1.1.2

Factor $20$ out of $- 20$ .

$20 (- 1) \frac{- y}{20} = - 20 \cdot - 12$

Step 4.6.1.1.1.3

Cancel the common factor.

$20 \cdot - 1 \frac{- y}{20} = - 20 \cdot - 12$

Step 4.6.1.1.1.4

Rewrite the expression.

$- - y = - 20 \cdot - 12$

Step 4.6.1.1.2

Multiply.

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Step 4.6.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 20 \cdot - 12$

Step 4.6.1.1.2.2

Multiply $y$ by $1$ .

$y = - 20 \cdot - 12$

Step 4.6.2

Simplify the right side.

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Step 4.6.2.1

Multiply $- 20$ by $- 12$ .

$y = 240$

Step 5

Solve $\frac{3}{4} y + 5 = - (\frac{4}{5} y - 7)$ for $y$ .

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Step 5.1

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

$\frac{3 y}{4} + 5 = - (\frac{4}{5} y - 7)$

Step 5.2

Simplify $- (\frac{4}{5} y - 7)$ .

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Step 5.2.1

Combine $\frac{4}{5}$ and $y$ .

$\frac{3 y}{4} + 5 = - (\frac{4 y}{5} - 7)$

Step 5.2.2

Apply the distributive property.

$\frac{3 y}{4} + 5 = - \frac{4 y}{5} - - 7$

Step 5.2.3

Multiply $- 1$ by $- 7$ .

$\frac{3 y}{4} + 5 = - \frac{4 y}{5} + 7$

Step 5.3

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

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Step 5.3.1

Add $\frac{4 y}{5}$ to both sides of the equation.

$\frac{3 y}{4} + 5 + \frac{4 y}{5} = 7$

Step 5.3.2

To write $\frac{3 y}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 y}{4} \cdot \frac{5}{5} + \frac{4 y}{5} + 5 = 7$

Step 5.3.3

To write $\frac{4 y}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 y}{4} \cdot \frac{5}{5} + \frac{4 y}{5} \cdot \frac{4}{4} + 5 = 7$

Step 5.3.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.4.1

Multiply $\frac{3 y}{4}$ by $\frac{5}{5}$ .

$\frac{3 y \cdot 5}{4 \cdot 5} + \frac{4 y}{5} \cdot \frac{4}{4} + 5 = 7$

Step 5.3.4.2

Multiply $4$ by $5$ .

$\frac{3 y \cdot 5}{20} + \frac{4 y}{5} \cdot \frac{4}{4} + 5 = 7$

Step 5.3.4.3

Multiply $\frac{4 y}{5}$ by $\frac{4}{4}$ .

$\frac{3 y \cdot 5}{20} + \frac{4 y \cdot 4}{5 \cdot 4} + 5 = 7$

Step 5.3.4.4

Multiply $5$ by $4$ .

$\frac{3 y \cdot 5}{20} + \frac{4 y \cdot 4}{20} + 5 = 7$

Step 5.3.5

Combine the numerators over the common denominator.

$\frac{3 y \cdot 5 + 4 y \cdot 4}{20} + 5 = 7$

Step 5.3.6

Simplify each term.

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Step 5.3.6.1

Simplify the numerator.

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Step 5.3.6.1.1

Factor $y$ out of $3 y \cdot 5 + 4 y \cdot 4$ .

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Step 5.3.6.1.1.1

Factor $y$ out of $3 y \cdot 5$ .

$\frac{y (3 \cdot 5) + 4 y \cdot 4}{20} + 5 = 7$

Step 5.3.6.1.1.2

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

$\frac{y (3 \cdot 5) + y (4 \cdot 4)}{20} + 5 = 7$

Step 5.3.6.1.1.3

Factor $y$ out of $y (3 \cdot 5) + y (4 \cdot 4)$ .

$\frac{y (3 \cdot 5 + 4 \cdot 4)}{20} + 5 = 7$

Step 5.3.6.1.2

Multiply $3$ by $5$ .

$\frac{y (15 + 4 \cdot 4)}{20} + 5 = 7$

Step 5.3.6.1.3

Multiply $4$ by $4$ .

$\frac{y (15 + 16)}{20} + 5 = 7$

Step 5.3.6.1.4

Add $15$ and $16$ .

$\frac{y \cdot 31}{20} + 5 = 7$

Step 5.3.6.2

Move $31$ to the left of $y$ .

$\frac{31 y}{20} + 5 = 7$

Step 5.4

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

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Step 5.4.1

Subtract $5$ from both sides of the equation.

$\frac{31 y}{20} = 7 - 5$

Step 5.4.2

Subtract $5$ from $7$ .

$\frac{31 y}{20} = 2$

Step 5.5

Multiply both sides of the equation by $\frac{20}{31}$ .

$\frac{20}{31} \cdot \frac{31 y}{20} = \frac{20}{31} \cdot 2$

Step 5.6

Simplify both sides of the equation.

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Step 5.6.1

Simplify the left side.

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Step 5.6.1.1

Simplify $\frac{20}{31} \cdot \frac{31 y}{20}$ .

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Step 5.6.1.1.1

Cancel the common factor of $20$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

$\frac{20}{31} \cdot \frac{31 y}{20} = \frac{20}{31} \cdot 2$

Step 5.6.1.1.1.2

Rewrite the expression.

$\frac{1}{31} (31 y) = \frac{20}{31} \cdot 2$

Step 5.6.1.1.2

Cancel the common factor of $31$ .

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Step 5.6.1.1.2.1

Factor $31$ out of $31 y$ .

$\frac{1}{31} (31 (y)) = \frac{20}{31} \cdot 2$

Step 5.6.1.1.2.2

Cancel the common factor.

$\frac{1}{31} (31 y) = \frac{20}{31} \cdot 2$

Step 5.6.1.1.2.3

Rewrite the expression.

$y = \frac{20}{31} \cdot 2$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Multiply $\frac{20}{31} \cdot 2$ .

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Step 5.6.2.1.1

Combine $\frac{20}{31}$ and $2$ .

$y = \frac{20 \cdot 2}{31}$

Step 5.6.2.1.2

Multiply $20$ by $2$ .

$y = \frac{40}{31}$

Step 6

List all of the solutions.

$y = 240, \frac{40}{31}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 240, \frac{40}{31}$

Decimal Form:

$y = 240, 1.29032258 \dots$

Mixed Number Form:

$y = 240, 1 \frac{9}{31}$