Algebra Examples

$\frac{5 x}{6 x + 3}$

Interchange the variables.

$x = \frac{5 y}{6 y + 3}$

Solve for $y$ .

Tap for more steps...

Rewrite the equation as $\frac{5 y}{6 y + 3} = x$ .

$\frac{5 y}{6 y + 3} = x$

Factor $3$ out of $6 y + 3$ .

Tap for more steps...

Factor $3$ out of $6 y$ .

$\frac{5 y}{3 (2 y) + 3} = x$

Factor $3$ out of $3$ .

$\frac{5 y}{3 (2 y) + 3 (1)} = x$

Factor $3$ out of $3 (2 y) + 3 (1)$ .

$\frac{5 y}{3 (2 y + 1)} = x$

Find the LCD of the terms in the equation.

Tap for more steps...

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

$3 (2 y + 1), 1$

The LCM of one and any expression is the expression.

$3 (2 y + 1)$

Multiply each term in $\frac{5 y}{3 (2 y + 1)} = x$ by $3 (2 y + 1)$ to eliminate the fractions.

Tap for more steps...

Multiply each term in $\frac{5 y}{3 (2 y + 1)} = x$ by $3 (2 y + 1)$ .

$\frac{5 y}{3 (2 y + 1)} (3 (2 y + 1)) = x (3 (2 y + 1))$

Simplify the left side.

Tap for more steps...

Rewrite using the commutative property of multiplication.

$3 \frac{5 y}{3 (2 y + 1)} (2 y + 1) = x (3 (2 y + 1))$

Cancel the common factor of $3$ .

Tap for more steps...

Cancel the common factor.

$3 \frac{5 y}{3 (2 y + 1)} (2 y + 1) = x (3 (2 y + 1))$

Rewrite the expression.

$\frac{5 y}{2 y + 1} (2 y + 1) = x (3 (2 y + 1))$

Cancel the common factor of $2 y + 1$ .

Tap for more steps...

Cancel the common factor.

$\frac{5 y}{2 y + 1} (2 y + 1) = x (3 (2 y + 1))$

Rewrite the expression.

$5 y = x (3 (2 y + 1))$

Simplify the right side.

Tap for more steps...

Rewrite using the commutative property of multiplication.

$5 y = 3 x (2 y + 1)$

Apply the distributive property.

$5 y = 3 x (2 y) + 3 x \cdot 1$

Simplify the expression.

Tap for more steps...

Rewrite using the commutative property of multiplication.

$5 y = 3 \cdot 2 x y + 3 x \cdot 1$

Multiply $3$ by $1$ .

$5 y = 3 \cdot 2 x y + 3 x$

Multiply $3$ by $2$ .

$5 y = 6 x y + 3 x$

Solve the equation.

Tap for more steps...

Subtract $6 x y$ from both sides of the equation.

$5 y - 6 x y = 3 x$

Factor $y$ out of $5 y - 6 x y$ .

Tap for more steps...

Factor $y$ out of $5 y$ .

$y \cdot 5 - 6 x y = 3 x$

Factor $y$ out of $- 6 x y$ .

$y \cdot 5 + y (- 6 x) = 3 x$

Factor $y$ out of $y \cdot 5 + y (- 6 x)$ .

$y (5 - 6 x) = 3 x$

Divide each term in $y (5 - 6 x) = 3 x$ by $5 - 6 x$ and simplify.

Tap for more steps...

Divide each term in $y (5 - 6 x) = 3 x$ by $5 - 6 x$ .

$\frac{y (5 - 6 x)}{5 - 6 x} = \frac{3 x}{5 - 6 x}$

Simplify the left side.

Tap for more steps...

Cancel the common factor of $5 - 6 x$ .

Tap for more steps...

Cancel the common factor.

$\frac{y (5 - 6 x)}{5 - 6 x} = \frac{3 x}{5 - 6 x}$

Divide $y$ by $1$ .

$y = \frac{3 x}{5 - 6 x}$

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

$f^{- 1} (x) = \frac{3 x}{5 - 6 x}$

Verify if $f^{- 1} (x) = \frac{3 x}{5 - 6 x}$ is the inverse of $f (x) = \frac{5 x}{6 x + 3}$ .

Tap for more steps...

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

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

Tap for more steps...

Set up the composite result function.

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

Evaluate $f^{- 1} (\frac{5 x}{6 x + 3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{6 x + 3})}{5 - 6 \frac{5 x}{6 x + 3}}$

Simplify terms.

Tap for more steps...

Factor $3$ out of $6 x + 3$ .

Tap for more steps...

Factor $3$ out of $6 x$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{3 (2 x) + 3})}{5 - 6 \frac{5 x}{6 x + 3}}$

Factor $3$ out of $3$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{3 (2 x) + 3 (1)})}{5 - 6 \frac{5 x}{6 x + 3}}$

Factor $3$ out of $3 (2 x) + 3 (1)$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{3 (2 x + 1)})}{5 - 6 \frac{5 x}{6 x + 3}}$

Factor $3$ out of $6 x + 3$ .

Tap for more steps...

Factor $3$ out of $6 x$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{3 (2 x + 1)})}{5 - 6 \frac{5 x}{3 (2 x) + 3}}$

Factor $3$ out of $3$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{3 (2 x + 1)})}{5 - 6 \frac{5 x}{3 (2 x) + 3 (1)}}$

Factor $3$ out of $3 (2 x) + 3 (1)$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{3 (\frac{5 x}{3 (2 x + 1)})}{5 - 6 \frac{5 x}{3 (2 x + 1)}}$

Combine $3$ and $\frac{5 x}{3 (2 x + 1)}$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 - 6 \frac{5 x}{3 (2 x + 1)}}$

Simplify the denominator.

Tap for more steps...

Cancel the common factor of $3$ .

Tap for more steps...

Factor $3$ out of $- 6$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 + 3 (- 2) (\frac{5 x}{3 (2 x + 1)})}$

Cancel the common factor.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 + 3 \cdot (- 2 \frac{5 x}{3 (2 x + 1)})}$

Rewrite the expression.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 - 2 \frac{5 x}{2 x + 1}}$

Combine $- 2$ and $\frac{5 x}{2 x + 1}$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 + \frac{- 2 (5 x)}{2 x + 1}}$

Multiply $5$ by $- 2$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 + \frac{- 10 x}{2 x + 1}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{5 - \frac{(10) x}{2 x + 1}}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{2 x + 1}{2 x + 1}$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5 (2 x + 1)}{2 x + 1} - \frac{10 x}{2 x + 1}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5 (2 x + 1) - 10 x}{2 x + 1}}$

Reorder terms.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{- 10 x + 5 (2 x + 1)}{2 x + 1}}$

Rewrite $\frac{- 10 x + 5 (2 x + 1)}{2 x + 1}$ in a factored form.

Tap for more steps...

Factor $5$ out of $- 10 x + 5 (2 x + 1)$ .

Tap for more steps...

Factor $5$ out of $- 10 x$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5 (- 2 x) + 5 (2 x + 1)}{2 x + 1}}$

Factor $5$ out of $5 (- 2 x) + 5 (2 x + 1)$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5 (- 2 x + 2 x + 1)}{2 x + 1}}$

Add $- 2 x$ and $2 x$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5 (0 + 1)}{2 x + 1}}$

Add $0$ and $1$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5 \cdot 1}{2 x + 1}}$

Multiply $5$ by $1$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5}{2 x + 1}}$

Multiply $3$ by $5$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{15 x}{3 (2 x + 1)}}{\frac{5}{2 x + 1}}$

Reduce the expression $\frac{15 x}{3 (2 x + 1)}$ by cancelling the common factors.

Tap for more steps...

Factor $3$ out of $15 x$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5}{2 x + 1}}$

Cancel the common factor.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{3 (5 x)}{3 (2 x + 1)}}{\frac{5}{2 x + 1}}$

Rewrite the expression.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{\frac{5 x}{2 x + 1}}{\frac{5}{2 x + 1}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{5 x}{2 x + 1} \cdot \frac{2 x + 1}{5}$

Cancel the common factor of $5$ .

Tap for more steps...

Factor $5$ out of $5 x$ .

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{5 (x)}{2 x + 1} \cdot \frac{2 x + 1}{5}$

Cancel the common factor.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{5 x}{2 x + 1} \cdot \frac{2 x + 1}{5}$

Rewrite the expression.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{x}{2 x + 1} \cdot (2 x + 1)$

Cancel the common factor of $2 x + 1$ .

Tap for more steps...

Cancel the common factor.

$f^{- 1} (\frac{5 x}{6 x + 3}) = \frac{x}{2 x + 1} \cdot (2 x + 1)$

Rewrite the expression.

$f^{- 1} (\frac{5 x}{6 x + 3}) = x$

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

Tap for more steps...

Set up the composite result function.

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

Evaluate $f (\frac{3 x}{5 - 6 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{5 (\frac{3 x}{5 - 6 x})}{6 (\frac{3 x}{5 - 6 x}) + 3}$

Combine $5$ and $\frac{3 x}{5 - 6 x}$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{6 (\frac{3 x}{5 - 6 x}) + 3}$

Simplify the denominator.

Tap for more steps...

Factor $3$ out of $6 \frac{3 x}{5 - 6 x} + 3$ .

Tap for more steps...

Factor $3$ out of $6 \frac{3 x}{5 - 6 x}$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (2 (\frac{3 x}{5 - 6 x})) + 3}$

Factor $3$ out of $3$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (2 (\frac{3 x}{5 - 6 x})) + 3 (1)}$

Factor $3$ out of $3 (2 \frac{3 x}{5 - 6 x}) + 3 (1)$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (2 (\frac{3 x}{5 - 6 x}) + 1)}$

Multiply $2 \frac{3 x}{5 - 6 x}$ .

Tap for more steps...

Combine $2$ and $\frac{3 x}{5 - 6 x}$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{2 (3 x)}{5 - 6 x} + 1)}$

Multiply $3$ by $2$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{6 x}{5 - 6 x} + 1)}$

Write $1$ as a fraction with a common denominator.

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{6 x}{5 - 6 x} + \frac{5 - 6 x}{5 - 6 x})}$

Combine the numerators over the common denominator.

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{6 x + 5 - 6 x}{5 - 6 x})}$

Reorder terms.

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{6 x - 6 x + 5}{- 6 x + 5})}$

Rewrite $\frac{6 x - 6 x + 5}{- 6 x + 5}$ in a factored form.

Tap for more steps...

Subtract $6 x$ from $6 x$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{0 + 5}{- 6 x + 5})}$

Add $0$ and $5$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{5 (3 x)}{5 - 6 x}}{3 (\frac{5}{- 6 x + 5})}$

Multiply $5$ by $3$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{15 x}{5 - 6 x}}{3 (\frac{5}{- 6 x + 5})}$

Combine fractions.

Tap for more steps...

Combine $3$ and $\frac{5}{- 6 x + 5}$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{15 x}{5 - 6 x}}{\frac{3 \cdot 5}{- 6 x + 5}}$

Multiply $3$ by $5$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{\frac{15 x}{5 - 6 x}}{\frac{15}{- 6 x + 5}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3 x}{5 - 6 x}) = \frac{15 x}{5 - 6 x} \cdot \frac{- 6 x + 5}{15}$

Cancel the common factor of $15$ .

Tap for more steps...

Factor $15$ out of $15 x$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{15 (x)}{5 - 6 x} \cdot \frac{- 6 x + 5}{15}$

Cancel the common factor.

$f (\frac{3 x}{5 - 6 x}) = \frac{15 x}{5 - 6 x} \cdot \frac{- 6 x + 5}{15}$

Rewrite the expression.

$f (\frac{3 x}{5 - 6 x}) = \frac{x}{5 - 6 x} \cdot (- 6 x + 5)$

Multiply $\frac{x}{5 - 6 x}$ by $- 6 x + 5$ .

$f (\frac{3 x}{5 - 6 x}) = \frac{x (- 6 x + 5)}{5 - 6 x}$

Cancel the common factor of $- 6 x + 5$ and $5 - 6 x$ .

Tap for more steps...

Reorder terms.

$f (\frac{3 x}{5 - 6 x}) = \frac{x (- 6 x + 5)}{- 6 x + 5}$

Cancel the common factor.

$f (\frac{3 x}{5 - 6 x}) = \frac{x (- 6 x + 5)}{- 6 x + 5}$

Divide $x$ by $1$ .

$f (\frac{3 x}{5 - 6 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{5 - 6 x}$ is the inverse of $f (x) = \frac{5 x}{6 x + 3}$ .

$f^{- 1} (x) = \frac{3 x}{5 - 6 x}$