Trigonometry Examples

$3 x + 2 y = 31$

Step 1

Subtract $3 x$ from both sides of the equation.

$2 y = 31 - 3 x$

Step 2

Divide each term in $2 y = 31 - 3 x$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 y = 31 - 3 x$ by $2$ .

$\frac{2 y}{2} = \frac{31}{2} + \frac{- 3 x}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{31}{2} + \frac{- 3 x}{2}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{31}{2} + \frac{- 3 x}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y = \frac{31}{2} - \frac{3 x}{2}$

Step 3

Interchange the variables.

$x = \frac{31}{2} - \frac{3 y}{2}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $\frac{31}{2} - \frac{3 y}{2} = x$ .

$\frac{31}{2} - \frac{3 y}{2} = x$

Step 4.2

Subtract $\frac{31}{2}$ from both sides of the equation.

$- \frac{3 y}{2} = x - \frac{31}{2}$

Step 4.3

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

$- \frac{2}{3} (- \frac{3 y}{2}) = - \frac{2}{3} (x - \frac{31}{2})$

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 $- \frac{2}{3} (- \frac{3 y}{2})$ .

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Step 4.4.1.1.1

Cancel the common factor of $2$ .

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Step 4.4.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 y}{2}) = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.1.2

Move the leading negative in $- \frac{3 y}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 y}{2} = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 y}{2} = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 y}{2} = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.2

Cancel the common factor of $3$ .

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Step 4.4.1.1.2.1

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.2.3

Rewrite the expression.

$- - y = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.3

Multiply.

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{2}{3} (x - \frac{31}{2})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- \frac{2}{3} (x - \frac{31}{2})$ .

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Step 4.4.2.1.1

Simplify terms.

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Step 4.4.2.1.1.1

Apply the distributive property.

$y = - \frac{2}{3} x - \frac{2}{3} (- \frac{31}{2})$

Step 4.4.2.1.1.2

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

$y = - \frac{x \cdot 2}{3} - \frac{2}{3} (- \frac{31}{2})$

Step 4.4.2.1.1.3

Cancel the common factor of $2$ .

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Step 4.4.2.1.1.3.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - \frac{x \cdot 2}{3} + \frac{- 2}{3} (- \frac{31}{2})$

Step 4.4.2.1.1.3.2

Move the leading negative in $- \frac{31}{2}$ into the numerator.

$y = - \frac{x \cdot 2}{3} + \frac{- 2}{3} \cdot \frac{- 31}{2}$

Step 4.4.2.1.1.3.3

Factor $2$ out of $- 2$ .

$y = - \frac{x \cdot 2}{3} + \frac{2 (- 1)}{3} \cdot \frac{- 31}{2}$

Step 4.4.2.1.1.3.4

Cancel the common factor.

$y = - \frac{x \cdot 2}{3} + \frac{2 \cdot - 1}{3} \cdot \frac{- 31}{2}$

Step 4.4.2.1.1.3.5

Rewrite the expression.

$y = - \frac{x \cdot 2}{3} + \frac{- 1}{3} \cdot - 31$

Step 4.4.2.1.1.4

Combine $\frac{- 1}{3}$ and $- 31$ .

$y = - \frac{x \cdot 2}{3} + \frac{- - 31}{3}$

Step 4.4.2.1.1.5

Multiply $- 1$ by $- 31$ .

$y = - \frac{x \cdot 2}{3} + \frac{31}{3}$

Step 4.4.2.1.2

Move $2$ to the left of $x$ .

$y = - \frac{2 x}{3} + \frac{31}{3}$

Step 5

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{2 x}{3} + \frac{31}{3}$

Step 6

Verify if $f^{- 1} (x) = - \frac{2 x}{3} + \frac{31}{3}$ is the inverse of $f (x) = \frac{31}{2} - \frac{3 x}{2}$ .

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Step 6.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 6.2

Evaluate $f^{- 1} (f (x))$ .

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Step 6.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 6.2.2

Evaluate $f^{- 1} (\frac{31}{2} - \frac{3 x}{2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = - \frac{2 (\frac{31}{2} - \frac{3 x}{2})}{3} + \frac{31}{3}$

Step 6.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 2 (\frac{31}{2} - \frac{3 x}{2}) + 31}{3}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Apply the distributive property.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 2 (\frac{31}{2}) - 2 (- \frac{3 x}{2}) + 31}{3}$

Step 6.2.4.2

Cancel the common factor of $2$ .

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Step 6.2.4.2.1

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{2 (- 1) (\frac{31}{2}) - 2 (- \frac{3 x}{2}) + 31}{3}$

Step 6.2.4.2.2

Cancel the common factor.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{2 \cdot (- 1 (\frac{31}{2})) - 2 (- \frac{3 x}{2}) + 31}{3}$

Step 6.2.4.2.3

Rewrite the expression.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 1 \cdot 31 - 2 (- \frac{3 x}{2}) + 31}{3}$

Step 6.2.4.3

Multiply $- 1$ by $31$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 31 - 2 (- \frac{3 x}{2}) + 31}{3}$

Step 6.2.4.4

Cancel the common factor of $2$ .

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Step 6.2.4.4.1

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 31 - 2 \frac{- 3 x}{2} + 31}{3}$

Step 6.2.4.4.2

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 31 + 2 (- 1) (\frac{- 3 x}{2}) + 31}{3}$

Step 6.2.4.4.3

Cancel the common factor.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 31 + 2 \cdot (- 1 \frac{- 3 x}{2}) + 31}{3}$

Step 6.2.4.4.4

Rewrite the expression.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 31 - 1 (- 3 x) + 31}{3}$

Step 6.2.4.5

Multiply $- 3$ by $- 1$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{- 31 + 3 x + 31}{3}$

Step 6.2.5

Simplify terms.

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Step 6.2.5.1

Combine the opposite terms in $- 31 + 3 x + 31$ .

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Step 6.2.5.1.1

Add $- 31$ and $31$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{3 x + 0}{3}$

Step 6.2.5.1.2

Add $3 x$ and $0$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{3 x}{3}$

Step 6.2.5.2

Cancel the common factor of $3$ .

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Step 6.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = \frac{3 x}{3}$

Step 6.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{31}{2} - \frac{3 x}{2}) = x$

Step 6.3

Evaluate $f (f^{- 1} (x))$ .

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Step 6.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 6.3.2

Evaluate $f (- \frac{2 x}{3} + \frac{31}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31}{2} - \frac{3 (- \frac{2 x}{3} + \frac{31}{3})}{2}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 - 3 (- \frac{2 x}{3} + \frac{31}{3})}{2}$

Step 6.3.4

Simplify each term.

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Step 6.3.4.1

Apply the distributive property.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 - 3 (- \frac{2 x}{3}) - 3 (\frac{31}{3})}{2}$

Step 6.3.4.2

Cancel the common factor of $3$ .

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Step 6.3.4.2.1

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 - 3 \frac{- 2 x}{3} - 3 (\frac{31}{3})}{2}$

Step 6.3.4.2.2

Factor $3$ out of $- 3$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 3 (- 1) (\frac{- 2 x}{3}) - 3 (\frac{31}{3})}{2}$

Step 6.3.4.2.3

Cancel the common factor.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 3 \cdot (- 1 \frac{- 2 x}{3}) - 3 (\frac{31}{3})}{2}$

Step 6.3.4.2.4

Rewrite the expression.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 - 1 (- 2 x) - 3 (\frac{31}{3})}{2}$

Step 6.3.4.3

Multiply $- 2$ by $- 1$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 2 x - 3 (\frac{31}{3})}{2}$

Step 6.3.4.4

Cancel the common factor of $3$ .

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Step 6.3.4.4.1

Factor $3$ out of $- 3$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 2 x + 3 (- 1) (\frac{31}{3})}{2}$

Step 6.3.4.4.2

Cancel the common factor.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 2 x + 3 \cdot (- 1 (\frac{31}{3}))}{2}$

Step 6.3.4.4.3

Rewrite the expression.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 2 x - 1 \cdot 31}{2}$

Step 6.3.4.5

Multiply $- 1$ by $31$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{31 + 2 x - 31}{2}$

Step 6.3.5

Simplify terms.

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Step 6.3.5.1

Combine the opposite terms in $31 + 2 x - 31$ .

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Step 6.3.5.1.1

Subtract $31$ from $31$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{2 x + 0}{2}$

Step 6.3.5.1.2

Add $2 x$ and $0$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{2 x}{2}$

Step 6.3.5.2

Cancel the common factor of $2$ .

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Step 6.3.5.2.1

Cancel the common factor.

$f (- \frac{2 x}{3} + \frac{31}{3}) = \frac{2 x}{2}$

Step 6.3.5.2.2

Divide $x$ by $1$ .

$f (- \frac{2 x}{3} + \frac{31}{3}) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2 x}{3} + \frac{31}{3}$ is the inverse of $f (x) = \frac{31}{2} - \frac{3 x}{2}$ .

$f^{- 1} (x) = - \frac{2 x}{3} + \frac{31}{3}$