Trigonometry Examples

$2 x + 3 y = 7$

Step 1

Subtract $2 x$ from both sides of the equation.

$3 y = 7 - 2 x$

Step 2

Divide each term in $3 y = 7 - 2 x$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 y = 7 - 2 x$ by $3$ .

$\frac{3 y}{3} = \frac{7}{3} + \frac{- 2 x}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{7}{3} + \frac{- 2 x}{3}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{3} + \frac{- 2 x}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y = \frac{7}{3} - \frac{2 x}{3}$

Step 3

Interchange the variables.

$x = \frac{7}{3} - \frac{2 y}{3}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $\frac{7}{3} - \frac{2 y}{3} = x$ .

$\frac{7}{3} - \frac{2 y}{3} = x$

Step 4.2

Subtract $\frac{7}{3}$ from both sides of the equation.

$- \frac{2 y}{3} = x - \frac{7}{3}$

Step 4.3

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

$- \frac{3}{2} (- \frac{2 y}{3}) = - \frac{3}{2} (x - \frac{7}{3})$

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

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Step 4.4.1.1.1

Cancel the common factor of $3$ .

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Step 4.4.1.1.1.1

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

$\frac{- 3}{2} (- \frac{2 y}{3}) = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.1.2

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

$\frac{- 3}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 y) = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Factor $2$ out of $- 2 y$ .

$\frac{- 1}{2} (2 (- y)) = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- y)) = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.2.3

Rewrite the expression.

$- - y = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.3

Multiply.

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{3}{2} (x - \frac{7}{3})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- \frac{3}{2} (x - \frac{7}{3})$ .

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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.

$y = - \frac{3}{2} x - \frac{3}{2} (- \frac{7}{3})$

Step 4.4.2.1.1.2

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

$y = - \frac{x \cdot 3}{2} - \frac{3}{2} (- \frac{7}{3})$

Step 4.4.2.1.1.3

Cancel the common factor of $3$ .

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Step 4.4.2.1.1.3.1

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

$y = - \frac{x \cdot 3}{2} + \frac{- 3}{2} (- \frac{7}{3})$

Step 4.4.2.1.1.3.2

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$y = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot \frac{- 7}{3}$

Step 4.4.2.1.1.3.3

Factor $3$ out of $- 3$ .

$y = - \frac{x \cdot 3}{2} + \frac{3 (- 1)}{2} \cdot \frac{- 7}{3}$

Step 4.4.2.1.1.3.4

Cancel the common factor.

$y = - \frac{x \cdot 3}{2} + \frac{3 \cdot - 1}{2} \cdot \frac{- 7}{3}$

Step 4.4.2.1.1.3.5

Rewrite the expression.

$y = - \frac{x \cdot 3}{2} + \frac{- 1}{2} \cdot - 7$

Step 4.4.2.1.1.4

Combine $\frac{- 1}{2}$ and $- 7$ .

$y = - \frac{x \cdot 3}{2} + \frac{- - 7}{2}$

Step 4.4.2.1.1.5

Multiply $- 1$ by $- 7$ .

$y = - \frac{x \cdot 3}{2} + \frac{7}{2}$

Step 4.4.2.1.2

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

$y = - \frac{3 x}{2} + \frac{7}{2}$

Step 5

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

$f^{- 1} (x) = - \frac{3 x}{2} + \frac{7}{2}$

Step 6

Verify if $f^{- 1} (x) = - \frac{3 x}{2} + \frac{7}{2}$ is the inverse of $f (x) = \frac{7}{3} - \frac{2 x}{3}$ .

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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 $f^{- 1} (\frac{7}{3} - \frac{2 x}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = - \frac{3 (\frac{7}{3} - \frac{2 x}{3})}{2} + \frac{7}{2}$

Step 6.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 3 (\frac{7}{3} - \frac{2 x}{3}) + 7}{2}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Apply the distributive property.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 3 (\frac{7}{3}) - 3 (- \frac{2 x}{3}) + 7}{2}$

Step 6.2.4.2

Cancel the common factor of $3$ .

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Step 6.2.4.2.1

Factor $3$ out of $- 3$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{3 (- 1) (\frac{7}{3}) - 3 (- \frac{2 x}{3}) + 7}{2}$

Step 6.2.4.2.2

Cancel the common factor.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{3 \cdot (- 1 (\frac{7}{3})) - 3 (- \frac{2 x}{3}) + 7}{2}$

Step 6.2.4.2.3

Rewrite the expression.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 1 \cdot 7 - 3 (- \frac{2 x}{3}) + 7}{2}$

Step 6.2.4.3

Multiply $- 1$ by $7$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 7 - 3 (- \frac{2 x}{3}) + 7}{2}$

Step 6.2.4.4

Cancel the common factor of $3$ .

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Step 6.2.4.4.1

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

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 7 - 3 \frac{- 2 x}{3} + 7}{2}$

Step 6.2.4.4.2

Factor $3$ out of $- 3$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 7 + 3 (- 1) (\frac{- 2 x}{3}) + 7}{2}$

Step 6.2.4.4.3

Cancel the common factor.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 7 + 3 \cdot (- 1 \frac{- 2 x}{3}) + 7}{2}$

Step 6.2.4.4.4

Rewrite the expression.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 7 - 1 (- 2 x) + 7}{2}$

Step 6.2.4.5

Multiply $- 2$ by $- 1$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{- 7 + 2 x + 7}{2}$

Step 6.2.5

Simplify terms.

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Step 6.2.5.1

Combine the opposite terms in $- 7 + 2 x + 7$ .

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Step 6.2.5.1.1

Add $- 7$ and $7$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{2 x + 0}{2}$

Step 6.2.5.1.2

Add $2 x$ and $0$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{2 x}{2}$

Step 6.2.5.2

Cancel the common factor of $2$ .

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Step 6.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = \frac{2 x}{2}$

Step 6.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{7}{3} - \frac{2 x}{3}) = x$

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 $f (- \frac{3 x}{2} + \frac{7}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7}{3} - \frac{2 (- \frac{3 x}{2} + \frac{7}{2})}{3}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 - 2 (- \frac{3 x}{2} + \frac{7}{2})}{3}$

Step 6.3.4

Simplify each term.

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Step 6.3.4.1

Apply the distributive property.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 - 2 (- \frac{3 x}{2}) - 2 (\frac{7}{2})}{3}$

Step 6.3.4.2

Cancel the common factor of $2$ .

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Step 6.3.4.2.1

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

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 - 2 \frac{- 3 x}{2} - 2 (\frac{7}{2})}{3}$

Step 6.3.4.2.2

Factor $2$ out of $- 2$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 2 (- 1) (\frac{- 3 x}{2}) - 2 (\frac{7}{2})}{3}$

Step 6.3.4.2.3

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 2 \cdot (- 1 \frac{- 3 x}{2}) - 2 (\frac{7}{2})}{3}$

Step 6.3.4.2.4

Rewrite the expression.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 - 1 (- 3 x) - 2 (\frac{7}{2})}{3}$

Step 6.3.4.3

Multiply $- 3$ by $- 1$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 3 x - 2 (\frac{7}{2})}{3}$

Step 6.3.4.4

Cancel the common factor of $2$ .

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Step 6.3.4.4.1

Factor $2$ out of $- 2$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 3 x + 2 (- 1) (\frac{7}{2})}{3}$

Step 6.3.4.4.2

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 3 x + 2 \cdot (- 1 (\frac{7}{2}))}{3}$

Step 6.3.4.4.3

Rewrite the expression.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 3 x - 1 \cdot 7}{3}$

Step 6.3.4.5

Multiply $- 1$ by $7$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{7 + 3 x - 7}{3}$

Step 6.3.5

Simplify terms.

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Step 6.3.5.1

Combine the opposite terms in $7 + 3 x - 7$ .

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Step 6.3.5.1.1

Subtract $7$ from $7$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{3 x + 0}{3}$

Step 6.3.5.1.2

Add $3 x$ and $0$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{3 x}{3}$

Step 6.3.5.2

Cancel the common factor of $3$ .

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Step 6.3.5.2.1

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{7}{2}) = \frac{3 x}{3}$

Step 6.3.5.2.2

Divide $x$ by $1$ .

$f (- \frac{3 x}{2} + \frac{7}{2}) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{3 x}{2} + \frac{7}{2}$ is the inverse of $f (x) = \frac{7}{3} - \frac{2 x}{3}$ .

$f^{- 1} (x) = - \frac{3 x}{2} + \frac{7}{2}$