Trigonometry Examples

$x + y = 180$ ; $x = \frac{1}{4} y - 24$

Step 1

Subtract $y$ from both sides of the equation.

$x = 180 - y$

$x = \frac{1}{4} y - 24$

Step 2

Replace all occurrences of $x$ with $180 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x = 180 - y$ with $180 - y$ .

$180 - y = 180 - y$

$x = \frac{1}{4} y - 24$

Step 2.2

Simplify the left side.

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Step 2.2.1

Remove parentheses.

$180 - y = 180 - y$

$x = \frac{1}{4} y - 24$

$180 - y = 180 - y$

$x = \frac{1}{4} y - 24$

Step 2.3

Replace all occurrences of $x$ in $x = \frac{1}{4} y - 24$ with $180 - y$ .

$180 - y = \frac{1}{4} y - 24$

$180 - y = 180 - y$

Step 2.4

Simplify $180 - y = \frac{1}{4} y - 24$ .

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Remove parentheses.

$180 - y = \frac{1}{4} y - 24$

$180 - y = 180 - y$

$180 - y = \frac{1}{4} y - 24$

$180 - y = 180 - y$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Combine $\frac{1}{4}$ and $y$ .

$180 - y = \frac{y}{4} - 24$

$180 - y = 180 - y$

$180 - y = \frac{y}{4} - 24$

$180 - y = 180 - y$

$180 - y = \frac{y}{4} - 24$

$180 - y = 180 - y$

$180 - y = \frac{y}{4} - 24$

$180 - y = 180 - y$

Step 3

Remove any equations from the system that are always true.

$180 - y = \frac{y}{4} - 24$

Step 4

Solve for $y$ in $180 - y = \frac{y}{4} - 24$ .

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Step 4.1

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

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Step 4.1.1

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

$180 - y - \frac{y}{4} = - 24$

Step 4.1.2

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

$180 - y \cdot \frac{4}{4} - \frac{y}{4} = - 24$

Step 4.1.3

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

$180 + \frac{- y \cdot 4}{4} - \frac{y}{4} = - 24$

Step 4.1.4

Combine the numerators over the common denominator.

$180 + \frac{- y \cdot 4 - y}{4} = - 24$

Step 4.1.5

Simplify each term.

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Step 4.1.5.1

Simplify the numerator.

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Step 4.1.5.1.1

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

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Step 4.1.5.1.1.1

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

$180 + \frac{y (- 1 \cdot 4) - y}{4} = - 24$

Step 4.1.5.1.1.2

Factor $y$ out of $- y$ .

$180 + \frac{y (- 1 \cdot 4) + y \cdot - 1}{4} = - 24$

Step 4.1.5.1.1.3

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

$180 + \frac{y (- 1 \cdot 4 - 1)}{4} = - 24$

Step 4.1.5.1.2

Multiply $- 1$ by $4$ .

$180 + \frac{y (- 4 - 1)}{4} = - 24$

Step 4.1.5.1.3

Subtract $1$ from $- 4$ .

$180 + \frac{y \cdot - 5}{4} = - 24$

Step 4.1.5.2

Move $- 5$ to the left of $y$ .

$180 + \frac{- 5 \cdot y}{4} = - 24$

Step 4.1.5.3

Move the negative in front of the fraction.

$180 - \frac{5 y}{4} = - 24$

Step 4.2

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

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Step 4.2.1

Subtract $180$ from both sides of the equation.

$- \frac{5 y}{4} = - 24 - 180$

Step 4.2.2

Subtract $180$ from $- 24$ .

$- \frac{5 y}{4} = - 204$

Step 4.3

Multiply both sides of the equation by $- \frac{4}{5}$ .

$- \frac{4}{5} (- \frac{5 y}{4}) = - \frac{4}{5} \cdot - 204$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Cancel the common factor of $4$ .

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Step 4.4.1.1.1.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$\frac{- 4}{5} (- \frac{5 y}{4}) = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.1.2

Move the leading negative in $- \frac{5 y}{4}$ into the numerator.

$\frac{- 4}{5} \cdot \frac{- 5 y}{4} = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{5} \cdot \frac{- 5 y}{4} = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{5} \cdot \frac{- 5 y}{4} = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{5} (- 5 y) = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.2

Cancel the common factor of $5$ .

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Step 4.4.1.1.2.1

Factor $5$ out of $- 5 y$ .

$\frac{- 1}{5} (5 (- y)) = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{5} (5 (- y)) = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.2.3

Rewrite the expression.

$- - y = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.3

Multiply.

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{4}{5} \cdot - 204$

Step 4.4.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{4}{5} \cdot - 204$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Multiply $- \frac{4}{5} \cdot - 204$ .

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Step 4.4.2.1.1

Multiply $- 204$ by $- 1$ .

$y = 204 (\frac{4}{5})$

Step 4.4.2.1.2

Combine $204$ and $\frac{4}{5}$ .

$y = \frac{204 \cdot 4}{5}$

Step 4.4.2.1.3

Multiply $204$ by $4$ .

$y = \frac{816}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{816}{5}$

Decimal Form:

$y = 163.2$

Mixed Number Form:

$y = 163 \frac{1}{5}$

Step 6