Trigonometry Examples

$y x + 3 y - 5 x = 0$

Step 1

Add $5 x$ to both sides of the equation.

$y x + 3 y = 5 x$

Step 2

Factor $y$ out of $y x + 3 y$ .

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Step 2.1

Factor $y$ out of $y x$ .

$y (x) + 3 y = 5 x$

Step 2.2

Factor $y$ out of $3 y$ .

$y (x) + y \cdot 3 = 5 x$

Step 2.3

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

$y (x + 3) = 5 x$

Step 3

Divide each term in $y (x + 3) = 5 x$ by $x + 3$ and simplify.

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Step 3.1

Divide each term in $y (x + 3) = 5 x$ by $x + 3$ .

$\frac{y (x + 3)}{x + 3} = \frac{5 x}{x + 3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x + 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{y (x + 3)}{x + 3} = \frac{5 x}{x + 3}$

Step 3.2.1.2

Divide $y$ by $1$ .

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

Step 4

Interchange the variables.

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

Step 5

Solve for $y$ .

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Step 5.1

Rewrite the equation as $\frac{5 y}{y + 3} = x$ .

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

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 3, 1$

Step 5.2.2

Remove parentheses.

$y + 3, 1$

Step 5.2.3

The LCM of one and any expression is the expression.

$y + 3$

Step 5.3

Multiply each term in $\frac{5 y}{y + 3} = x$ by $y + 3$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{5 y}{y + 3} = x$ by $y + 3$ .

$\frac{5 y}{y + 3} (y + 3) = x (y + 3)$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $y + 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{5 y}{y + 3} (y + 3) = x (y + 3)$

Step 5.3.2.1.2

Rewrite the expression.

$5 y = x (y + 3)$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Apply the distributive property.

$5 y = x y + x \cdot 3$

Step 5.3.3.2

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

$5 y = x y + 3 x$

Step 5.4

Solve the equation.

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Step 5.4.1

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

$5 y - x y = 3 x$

Step 5.4.2

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

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Step 5.4.2.1

Factor $y$ out of $5 y$ .

$y \cdot 5 - x y = 3 x$

Step 5.4.2.2

Factor $y$ out of $- x y$ .

$y \cdot 5 + y (- x) = 3 x$

Step 5.4.2.3

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

$y (5 - x) = 3 x$

Step 5.4.3

Divide each term in $y (5 - x) = 3 x$ by $5 - x$ and simplify.

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Step 5.4.3.1

Divide each term in $y (5 - x) = 3 x$ by $5 - x$ .

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

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $5 - x$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

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

Step 5.4.3.2.1.2

Divide $y$ by $1$ .

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

Step 6

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

$f^{- 1} (x) = \frac{3 x}{5 - x}$

Step 7

Verify if $f^{- 1} (x) = \frac{3 x}{5 - x}$ is the inverse of $f (x) = \frac{5 x}{x + 3}$ .

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Step 7.1

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

Step 7.2

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

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Step 7.2.1

Set up the composite result function.

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

Step 7.2.2

Evaluate $f^{- 1} (\frac{5 x}{x + 3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{3 (\frac{5 x}{x + 3})}{5 - (\frac{5 x}{x + 3})}$

Step 7.2.3

Combine $3$ and $\frac{5 x}{x + 3}$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{5 - \frac{5 x}{x + 3}}$

Step 7.2.4

Simplify the denominator.

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Step 7.2.4.1

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

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{5 (x + 3)}{x + 3} - \frac{5 x}{x + 3}}$

Step 7.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{5 (x + 3) - 5 x}{x + 3}}$

Step 7.2.4.3

Reorder terms.

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{- 5 x + 5 (x + 3)}{x + 3}}$

Step 7.2.4.4

Rewrite $\frac{- 5 x + 5 (x + 3)}{x + 3}$ in a factored form.

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Step 7.2.4.4.1

Factor $5$ out of $- 5 x + 5 (x + 3)$ .

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Step 7.2.4.4.1.1

Factor $5$ out of $- 5 x$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{5 (- x) + 5 (x + 3)}{x + 3}}$

Step 7.2.4.4.1.2

Factor $5$ out of $5 (- x) + 5 (x + 3)$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{5 (- x + x + 3)}{x + 3}}$

Step 7.2.4.4.2

Add $- x$ and $x$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{5 (0 + 3)}{x + 3}}$

Step 7.2.4.4.3

Add $0$ and $3$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{5 \cdot 3}{x + 3}}$

Step 7.2.4.5

Multiply $5$ by $3$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{3 (5 x)}{x + 3}}{\frac{15}{x + 3}}$

Step 7.2.5

Multiply $3$ by $5$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{\frac{15 x}{x + 3}}{\frac{15}{x + 3}}$

Step 7.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{15 x}{x + 3} \cdot \frac{x + 3}{15}$

Step 7.2.7

Cancel the common factor of $15$ .

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Step 7.2.7.1

Factor $15$ out of $15 x$ .

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{15 (x)}{x + 3} \cdot \frac{x + 3}{15}$

Step 7.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{15 x}{x + 3} \cdot \frac{x + 3}{15}$

Step 7.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{x}{x + 3} \cdot (x + 3)$

Step 7.2.8

Cancel the common factor of $x + 3$ .

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Step 7.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{5 x}{x + 3}) = \frac{x}{x + 3} \cdot (x + 3)$

Step 7.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{5 x}{x + 3}) = x$

Step 7.3

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

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Step 7.3.1

Set up the composite result function.

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

Step 7.3.2

Evaluate $f (\frac{3 x}{5 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{5 - x}) = \frac{5 (\frac{3 x}{5 - x})}{(\frac{3 x}{5 - x}) + 3}$

Step 7.3.3

Combine $5$ and $\frac{3 x}{5 - x}$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 x}{5 - x} + 3}$

Step 7.3.4

Simplify the denominator.

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Step 7.3.4.1

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

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 x}{5 - x} + \frac{3 (5 - x)}{5 - x}}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 x + 3 (5 - x)}{5 - x}}$

Step 7.3.4.3

Reorder terms.

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 x + 3 (5 - x)}{- x + 5}}$

Step 7.3.4.4

Rewrite $\frac{3 x + 3 (5 - x)}{- x + 5}$ in a factored form.

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Step 7.3.4.4.1

Factor $3$ out of $3 x + 3 (5 - x)$ .

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Step 7.3.4.4.1.1

Factor $3$ out of $3 x$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 (x) + 3 (5 - x)}{- x + 5}}$

Step 7.3.4.4.1.2

Factor $3$ out of $3 (x) + 3 (5 - x)$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 (x + 5 - x)}{- x + 5}}$

Step 7.3.4.4.2

Subtract $x$ from $x$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 (0 + 5)}{- x + 5}}$

Step 7.3.4.4.3

Add $0$ and $5$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{3 \cdot 5}{- x + 5}}$

Step 7.3.4.5

Multiply $3$ by $5$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{5 (3 x)}{5 - x}}{\frac{15}{- x + 5}}$

Step 7.3.5

Multiply $5$ by $3$ .

$f (\frac{3 x}{5 - x}) = \frac{\frac{15 x}{5 - x}}{\frac{15}{- x + 5}}$

Step 7.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3 x}{5 - x}) = \frac{15 x}{5 - x} \cdot \frac{- x + 5}{15}$

Step 7.3.7

Cancel the common factor of $15$ .

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Step 7.3.7.1

Factor $15$ out of $15 x$ .

$f (\frac{3 x}{5 - x}) = \frac{15 (x)}{5 - x} \cdot \frac{- x + 5}{15}$

Step 7.3.7.2

Cancel the common factor.

$f (\frac{3 x}{5 - x}) = \frac{15 x}{5 - x} \cdot \frac{- x + 5}{15}$

Step 7.3.7.3

Rewrite the expression.

$f (\frac{3 x}{5 - x}) = \frac{x}{5 - x} \cdot (- x + 5)$

Step 7.3.8

Multiply $\frac{x}{5 - x}$ by $- x + 5$ .

$f (\frac{3 x}{5 - x}) = \frac{x (- x + 5)}{5 - x}$

Step 7.3.9

Cancel the common factor of $- x + 5$ and $5 - x$ .

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Step 7.3.9.1

Reorder terms.

$f (\frac{3 x}{5 - x}) = \frac{x (- x + 5)}{- x + 5}$

Step 7.3.9.2

Cancel the common factor.

$f (\frac{3 x}{5 - x}) = \frac{x (- x + 5)}{- x + 5}$

Step 7.3.9.3

Divide $x$ by $1$ .

$f (\frac{3 x}{5 - x}) = x$

Step 7.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{5 - x}$ is the inverse of $f (x) = \frac{5 x}{x + 3}$ .

$f^{- 1} (x) = \frac{3 x}{5 - x}$