Trigonometry Examples

$- 4 x + 3 y = 25$

Step 1

Add $4 x$ to both sides of the equation.

$3 y = 25 + 4 x$

Step 2

Divide each term in $3 y = 25 + 4 x$ by $3$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $3 y = 25 + 4 x$ by $3$ .

$\frac{3 y}{3} = \frac{25}{3} + \frac{4 x}{3}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{25}{3} + \frac{4 x}{3}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{25}{3} + \frac{4 x}{3}$

Step 3

Interchange the variables.

$x = \frac{25}{3} + \frac{4 y}{3}$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $\frac{25}{3} + \frac{4 y}{3} = x$ .

$\frac{25}{3} + \frac{4 y}{3} = x$

Step 4.2

Subtract $\frac{25}{3}$ from both sides of the equation.

$\frac{4 y}{3} = x - \frac{25}{3}$

Step 4.3

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x - \frac{25}{3})$

Step 4.4

Simplify both sides of the equation.

Tap for more steps...

Step 4.4.1

Simplify the left side.

Tap for more steps...

Step 4.4.1.1

Simplify $\frac{3}{4} \cdot \frac{4 y}{3}$ .

Tap for more steps...

Step 4.4.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x - \frac{25}{3})$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 y) = \frac{3}{4} (x - \frac{25}{3})$

Step 4.4.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.4.1.1.2.1

Factor $4$ out of $4 y$ .

$\frac{1}{4} (4 (y)) = \frac{3}{4} (x - \frac{25}{3})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 y) = \frac{3}{4} (x - \frac{25}{3})$

Step 4.4.1.1.2.3

Rewrite the expression.

$y = \frac{3}{4} (x - \frac{25}{3})$

Step 4.4.2

Simplify the right side.

Tap for more steps...

Step 4.4.2.1

Simplify $\frac{3}{4} (x - \frac{25}{3})$ .

Tap for more steps...

Step 4.4.2.1.1

Apply the distributive property.

$y = \frac{3}{4} x + \frac{3}{4} (- \frac{25}{3})$

Step 4.4.2.1.2

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

$y = \frac{3 x}{4} + \frac{3}{4} (- \frac{25}{3})$

Step 4.4.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.4.2.1.3.1

Move the leading negative in $- \frac{25}{3}$ into the numerator.

$y = \frac{3 x}{4} + \frac{3}{4} \cdot \frac{- 25}{3}$

Step 4.4.2.1.3.2

Cancel the common factor.

$y = \frac{3 x}{4} + \frac{3}{4} \cdot \frac{- 25}{3}$

Step 4.4.2.1.3.3

Rewrite the expression.

$y = \frac{3 x}{4} + \frac{1}{4} \cdot - 25$

Step 4.4.2.1.4

Combine $\frac{1}{4}$ and $- 25$ .

$y = \frac{3 x}{4} + \frac{- 25}{4}$

Step 4.4.2.1.5

Move the negative in front of the fraction.

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 5

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

$f^{- 1} (x) = \frac{3 x}{4} - \frac{25}{4}$

Step 6

Verify if $f^{- 1} (x) = \frac{3 x}{4} - \frac{25}{4}$ is the inverse of $f (x) = \frac{25}{3} + \frac{4 x}{3}$ .

Tap for more steps...

Step 6.1

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

Step 6.2

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

Tap for more steps...

Step 6.2.1

Set up the composite result function.

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

Step 6.2.2

Evaluate $f^{- 1} (\frac{25}{3} + \frac{4 x}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{3 (\frac{25}{3} + \frac{4 x}{3})}{4} - \frac{25}{4}$

Step 6.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{3 (\frac{25}{3} + \frac{4 x}{3}) - 25}{4}$

Step 6.2.4

Simplify each term.

Tap for more steps...

Step 6.2.4.1

Apply the distributive property.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{3 (\frac{25}{3}) + 3 (\frac{4 x}{3}) - 25}{4}$

Step 6.2.4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.4.2.1

Cancel the common factor.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{3 (\frac{25}{3}) + 3 (\frac{4 x}{3}) - 25}{4}$

Step 6.2.4.2.2

Rewrite the expression.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{25 + 3 (\frac{4 x}{3}) - 25}{4}$

Step 6.2.4.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{25 + 3 (\frac{4 x}{3}) - 25}{4}$

Step 6.2.4.3.2

Rewrite the expression.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{25 + 4 x - 25}{4}$

Step 6.2.5

Simplify terms.

Tap for more steps...

Step 6.2.5.1

Combine the opposite terms in $25 + 4 x - 25$ .

Tap for more steps...

Step 6.2.5.1.1

Subtract $25$ from $25$ .

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{4 x + 0}{4}$

Step 6.2.5.1.2

Add $4 x$ and $0$ .

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{4 x}{4}$

Step 6.2.5.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = \frac{4 x}{4}$

Step 6.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{25}{3} + \frac{4 x}{3}) = x$

Step 6.3

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

Tap for more steps...

Step 6.3.1

Set up the composite result function.

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

Step 6.3.2

Evaluate $f (\frac{3 x}{4} - \frac{25}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25}{3} + \frac{4 (\frac{3 x}{4} - \frac{25}{4})}{3}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 4 (\frac{3 x}{4} - \frac{25}{4})}{3}$

Step 6.3.4

Simplify each term.

Tap for more steps...

Step 6.3.4.1

Apply the distributive property.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 4 (\frac{3 x}{4}) + 4 (- \frac{25}{4})}{3}$

Step 6.3.4.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.3.4.2.1

Cancel the common factor.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 4 (\frac{3 x}{4}) + 4 (- \frac{25}{4})}{3}$

Step 6.3.4.2.2

Rewrite the expression.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 3 x + 4 (- \frac{25}{4})}{3}$

Step 6.3.4.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.3.4.3.1

Move the leading negative in $- \frac{25}{4}$ into the numerator.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 3 x + 4 (\frac{- 25}{4})}{3}$

Step 6.3.4.3.2

Cancel the common factor.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 3 x + 4 (\frac{- 25}{4})}{3}$

Step 6.3.4.3.3

Rewrite the expression.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{25 + 3 x - 25}{3}$

Step 6.3.5

Simplify terms.

Tap for more steps...

Step 6.3.5.1

Combine the opposite terms in $25 + 3 x - 25$ .

Tap for more steps...

Step 6.3.5.1.1

Subtract $25$ from $25$ .

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{3 x + 0}{3}$

Step 6.3.5.1.2

Add $3 x$ and $0$ .

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{3 x}{3}$

Step 6.3.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.5.2.1

Cancel the common factor.

$f (\frac{3 x}{4} - \frac{25}{4}) = \frac{3 x}{3}$

Step 6.3.5.2.2

Divide $x$ by $1$ .

$f (\frac{3 x}{4} - \frac{25}{4}) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{4} - \frac{25}{4}$ is the inverse of $f (x) = \frac{25}{3} + \frac{4 x}{3}$ .

$f^{- 1} (x) = \frac{3 x}{4} - \frac{25}{4}$