Find the Inverse (6x-5)/(7x+3)

$\frac{6 x - 5}{7 x + 3}$

Step 1

Interchange the variables.

$x = \frac{6 y - 5}{7 y + 3}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{6 y - 5}{7 y + 3} = x$ .

$\frac{6 y - 5}{7 y + 3} = x$

Find the LCD of the terms in the equation.

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

$7 y + 3, 1$

Remove parentheses.

$7 y + 3, 1$

The LCM of one and any expression is the expression.

$7 y + 3$

Multiply each term in $\frac{6 y - 5}{7 y + 3} = x$ by $7 y + 3$ to eliminate the fractions.

Multiply each term in $\frac{6 y - 5}{7 y + 3} = x$ by $7 y + 3$ .

$\frac{6 y - 5}{7 y + 3} (7 y + 3) = x (7 y + 3)$

Simplify the left side.

Cancel the common factor of $7 y + 3$ .

Cancel the common factor.

$\frac{6 y - 5}{7 y + 3} (7 y + 3) = x (7 y + 3)$

Rewrite the expression.

$6 y - 5 = x (7 y + 3)$

Simplify the right side.

Apply the distributive property.

$6 y - 5 = x (7 y) + x \cdot 3$

Reorder.

Rewrite using the commutative property of multiplication.

$6 y - 5 = 7 x y + x \cdot 3$

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

$6 y - 5 = 7 x y + 3 x$

Solve the equation.

Subtract $7 x y$ from both sides of the equation.

$6 y - 5 - 7 x y = 3 x$

Add $5$ to both sides of the equation.

$6 y - 7 x y = 3 x + 5$

Factor $y$ out of $6 y - 7 x y$ .

Factor $y$ out of $6 y$ .

$y \cdot 6 - 7 x y = 3 x + 5$

Factor $y$ out of $- 7 x y$ .

$y \cdot 6 + y (- 7 x) = 3 x + 5$

Factor $y$ out of $y \cdot 6 + y (- 7 x)$ .

$y (6 - 7 x) = 3 x + 5$

Divide each term in $y (6 - 7 x) = 3 x + 5$ by $6 - 7 x$ and simplify.

Divide each term in $y (6 - 7 x) = 3 x + 5$ by $6 - 7 x$ .

$\frac{y (6 - 7 x)}{6 - 7 x} = \frac{3 x}{6 - 7 x} + \frac{5}{6 - 7 x}$

Simplify the left side.

Cancel the common factor of $6 - 7 x$ .

Cancel the common factor.

$\frac{y (6 - 7 x)}{6 - 7 x} = \frac{3 x}{6 - 7 x} + \frac{5}{6 - 7 x}$

Divide $y$ by $1$ .

$y = \frac{3 x}{6 - 7 x} + \frac{5}{6 - 7 x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{3 x + 5}{6 - 7 x}$

Step 3

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

$f^{- 1} (x) = \frac{3 x + 5}{6 - 7 x}$

Step 4

Verify if $f^{- 1} (x) = \frac{3 x + 5}{6 - 7 x}$ is the inverse of $f (x) = \frac{6 x - 5}{7 x + 3}$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\frac{6 x - 5}{7 x + 3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{3 (\frac{6 x - 5}{7 x + 3}) + 5}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Simplify the numerator.

Combine $3$ and $\frac{6 x - 5}{7 x + 3}$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{3 (6 x - 5)}{7 x + 3} + 5}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{7 x + 3}{7 x + 3}$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{3 (6 x - 5)}{7 x + 3} + \frac{5 (7 x + 3)}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{3 (6 x - 5) + 5 (7 x + 3)}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Reorder terms.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{5 (7 x + 3) + 3 (6 x - 5)}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Rewrite $\frac{5 (7 x + 3) + 3 (6 x - 5)}{7 x + 3}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{5 (7 x) + 5 \cdot 3 + 3 (6 x - 5)}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Multiply $7$ by $5$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{35 x + 5 \cdot 3 + 3 (6 x - 5)}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{35 x + 15 + 3 (6 x - 5)}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{35 x + 15 + 3 (6 x) + 3 \cdot - 5}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Multiply $6$ by $3$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{35 x + 15 + 18 x + 3 \cdot - 5}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Multiply $3$ by $- 5$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{35 x + 15 + 18 x - 15}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Add $35 x$ and $18 x$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x + 15 - 15}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Subtract $15$ from $15$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x + 0}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Add $53 x$ and $0$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{6 - 7 \frac{6 x - 5}{7 x + 3}}$

Simplify the denominator.

Combine $- 7$ and $\frac{6 x - 5}{7 x + 3}$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{6 + \frac{- 7 (6 x - 5)}{7 x + 3}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{6 - \frac{(7) (6 x - 5)}{7 x + 3}}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{7 x + 3}{7 x + 3}$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{6 (7 x + 3)}{7 x + 3} - \frac{7 (6 x - 5)}{7 x + 3}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{6 (7 x + 3) - 7 (6 x - 5)}{7 x + 3}}$

Rewrite $\frac{6 (7 x + 3) - 7 (6 x - 5)}{7 x + 3}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{6 (7 x) + 6 \cdot 3 - 7 (6 x - 5)}{7 x + 3}}$

Multiply $7$ by $6$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{42 x + 6 \cdot 3 - 7 (6 x - 5)}{7 x + 3}}$

Multiply $6$ by $3$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{42 x + 18 - 7 (6 x - 5)}{7 x + 3}}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{42 x + 18 - 7 (6 x) - 7 \cdot - 5}{7 x + 3}}$

Multiply $6$ by $- 7$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{42 x + 18 - 42 x - 7 \cdot - 5}{7 x + 3}}$

Multiply $- 7$ by $- 5$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{42 x + 18 - 42 x + 35}{7 x + 3}}$

Subtract $42 x$ from $42 x$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{0 + 18 + 35}{7 x + 3}}$

Add $0$ and $18$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{18 + 35}{7 x + 3}}$

Add $18$ and $35$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{\frac{53 x}{7 x + 3}}{\frac{53}{7 x + 3}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{53 x}{7 x + 3} \cdot \frac{7 x + 3}{53}$

Cancel the common factor of $53$ .

Factor $53$ out of $53 x$ .

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{53 (x)}{7 x + 3} \cdot \frac{7 x + 3}{53}$

Cancel the common factor.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{53 x}{7 x + 3} \cdot \frac{7 x + 3}{53}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{x}{7 x + 3} \cdot (7 x + 3)$

Cancel the common factor of $7 x + 3$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = \frac{x}{7 x + 3} \cdot (7 x + 3)$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 5}{7 x + 3}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{3 x + 5}{6 - 7 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{6 (\frac{3 x + 5}{6 - 7 x}) - 5}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Simplify the numerator.

Combine $6$ and $\frac{3 x + 5}{6 - 7 x}$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{6 (3 x + 5)}{6 - 7 x} - 5}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{6 - 7 x}{6 - 7 x}$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{6 (3 x + 5)}{6 - 7 x} - 5 \cdot \frac{6 - 7 x}{6 - 7 x}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Combine $- 5$ and $\frac{6 - 7 x}{6 - 7 x}$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{6 (3 x + 5)}{6 - 7 x} + \frac{- 5 (6 - 7 x)}{6 - 7 x}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Combine the numerators over the common denominator.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{6 (3 x + 5) - 5 (6 - 7 x)}{6 - 7 x}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Reorder terms.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{6 (3 x + 5) - 5 (6 - 7 x)}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Rewrite $\frac{6 (3 x + 5) - 5 (6 - 7 x)}{- 7 x + 6}$ in a factored form.

Apply the distributive property.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{6 (3 x) + 6 \cdot 5 - 5 (6 - 7 x)}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Multiply $3$ by $6$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{18 x + 6 \cdot 5 - 5 (6 - 7 x)}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Multiply $6$ by $5$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{18 x + 30 - 5 (6 - 7 x)}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Apply the distributive property.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{18 x + 30 - 5 \cdot 6 - 5 (- 7 x)}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Multiply $- 5$ by $6$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{18 x + 30 - 30 - 5 (- 7 x)}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Multiply $- 7$ by $- 5$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{18 x + 30 - 30 + 35 x}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Add $18 x$ and $35 x$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x + 30 - 30}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Subtract $30$ from $30$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x + 0}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Add $53 x$ and $0$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{7 (\frac{3 x + 5}{6 - 7 x}) + 3}$

Simplify the denominator.

Combine $7$ and $\frac{3 x + 5}{6 - 7 x}$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{7 (3 x + 5)}{6 - 7 x} + 3}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{6 - 7 x}{6 - 7 x}$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{7 (3 x + 5)}{6 - 7 x} + \frac{3 (6 - 7 x)}{6 - 7 x}}$

Combine the numerators over the common denominator.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{7 (3 x + 5) + 3 (6 - 7 x)}{6 - 7 x}}$

Reorder terms.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{7 (3 x + 5) + 3 (6 - 7 x)}{- 7 x + 6}}$

Rewrite $\frac{7 (3 x + 5) + 3 (6 - 7 x)}{- 7 x + 6}$ in a factored form.

Apply the distributive property.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{7 (3 x) + 7 \cdot 5 + 3 (6 - 7 x)}{- 7 x + 6}}$

Multiply $3$ by $7$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{21 x + 7 \cdot 5 + 3 (6 - 7 x)}{- 7 x + 6}}$

Multiply $7$ by $5$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{21 x + 35 + 3 (6 - 7 x)}{- 7 x + 6}}$

Apply the distributive property.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{21 x + 35 + 3 \cdot 6 + 3 (- 7 x)}{- 7 x + 6}}$

Multiply $3$ by $6$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{21 x + 35 + 18 + 3 (- 7 x)}{- 7 x + 6}}$

Multiply $- 7$ by $3$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{21 x + 35 + 18 - 21 x}{- 7 x + 6}}$

Subtract $21 x$ from $21 x$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{0 + 35 + 18}{- 7 x + 6}}$

Add $0$ and $35$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{35 + 18}{- 7 x + 6}}$

Add $35$ and $18$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{\frac{53 x}{- 7 x + 6}}{\frac{53}{- 7 x + 6}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{53 x}{- 7 x + 6} \cdot \frac{- 7 x + 6}{53}$

Cancel the common factor of $53$ .

Factor $53$ out of $53 x$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{53 (x)}{- 7 x + 6} \cdot \frac{- 7 x + 6}{53}$

Cancel the common factor.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{53 x}{- 7 x + 6} \cdot \frac{- 7 x + 6}{53}$

Rewrite the expression.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{x}{- 7 x + 6} \cdot (- 7 x + 6)$

Multiply $\frac{x}{- 7 x + 6}$ by $- 7 x + 6$ .

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{x (- 7 x + 6)}{- 7 x + 6}$

Cancel the common factor of $- 7 x + 6$ .

Cancel the common factor.

$f (\frac{3 x + 5}{6 - 7 x}) = \frac{x (- 7 x + 6)}{- 7 x + 6}$

Divide $x$ by $1$ .

$f (\frac{3 x + 5}{6 - 7 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x + 5}{6 - 7 x}$ is the inverse of $f (x) = \frac{6 x - 5}{7 x + 3}$ .

$f^{- 1} (x) = \frac{3 x + 5}{6 - 7 x}$

Find the Inverse (6x-5)/(7x+3)

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