Algebra Examples

$\frac{6 x}{7 x - 1}$

Step 1

Interchange the variables.

$x = \frac{6 y}{7 y - 1}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Multiply the equation by $7 y - 1$ .

$(7 y - 1) (x) = (\frac{6 y}{7 y - 1}) (7 y - 1)$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $(7 y - 1) (x)$ .

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$7 y x - 1 x = (\frac{6 y}{7 y - 1}) (7 y - 1)$

Step 2.2.1.2

Rewrite $- 1 x$ as $- x$ .

$7 y x - x = (\frac{6 y}{7 y - 1}) (7 y - 1)$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $7 y - 1$ .

Tap for more steps...

Step 2.3.1.1

Cancel the common factor.

$7 y x - x = \frac{6 y}{7 y - 1} (7 y - 1)$

Step 2.3.1.2

Rewrite the expression.

$7 y x - x = 6 y$

Step 2.4

Solve for $y$ .

Tap for more steps...

Step 2.4.1

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

$7 y x - x - 6 y = 0$

Step 2.4.2

Add $x$ to both sides of the equation.

$7 y x - 6 y = x$

Step 2.4.3

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

Tap for more steps...

Step 2.4.3.1

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

$y (7 x) - 6 y = x$

Step 2.4.3.2

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

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

Step 2.4.3.3

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

$y (7 x - 6) = x$

Step 2.4.4

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

Tap for more steps...

Step 2.4.4.1

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

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

Step 2.4.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.4.2.1

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

Tap for more steps...

Step 2.4.4.2.1.1

Cancel the common factor.

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

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{7 x - 6}$

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.4

Simplify the denominator.

Tap for more steps...

Step 4.2.4.1

Multiply $7 \frac{6 x}{7 x - 1}$ .

Tap for more steps...

Step 4.2.4.1.1

Combine $7$ and $\frac{6 x}{7 x - 1}$ .

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

Step 4.2.4.1.2

Multiply $6$ by $7$ .

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

Step 4.2.4.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{7 x - 1}{7 x - 1}$ .

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

Step 4.2.4.3

Combine $- 6$ and $\frac{7 x - 1}{7 x - 1}$ .

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

Step 4.2.4.4

Combine the numerators over the common denominator.

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

Step 4.2.4.5

Rewrite $\frac{42 x - 6 (7 x - 1)}{7 x - 1}$ in a factored form.

Tap for more steps...

Step 4.2.4.5.1

Factor $6$ out of $42 x - 6 (7 x - 1)$ .

Tap for more steps...

Step 4.2.4.5.1.1

Factor $6$ out of $42 x$ .

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

Step 4.2.4.5.1.2

Factor $6$ out of $- 6 (7 x - 1)$ .

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

Step 4.2.4.5.1.3

Factor $6$ out of $6 (7 x) + 6 (- (7 x - 1))$ .

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

Step 4.2.4.5.2

Apply the distributive property.

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

Step 4.2.4.5.3

Multiply $7$ by $- 1$ .

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

Step 4.2.4.5.4

Multiply $- 1$ by $- 1$ .

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

Step 4.2.4.5.5

Subtract $7 x$ from $7 x$ .

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

Step 4.2.4.5.6

Add $0$ and $1$ .

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

Step 4.2.4.6

Multiply $6$ by $1$ .

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

Step 4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.6

Multiply $\frac{7 x - 1}{6}$ by $1$ .

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

Step 4.2.7

Cancel the common factor of $6$ .

Tap for more steps...

Step 4.2.7.1

Factor $6$ out of $6 x$ .

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

Step 4.2.7.2

Cancel the common factor.

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

Step 4.2.7.3

Rewrite the expression.

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

Step 4.2.8

Cancel the common factor of $7 x - 1$ .

Tap for more steps...

Step 4.2.8.1

Cancel the common factor.

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

Step 4.2.8.2

Rewrite the expression.

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

$f (\frac{x}{7 x - 6}) = \frac{6 (\frac{x}{7 x - 6})}{7 (\frac{x}{7 x - 6}) - 1}$

Step 4.3.3

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

$f (\frac{x}{7 x - 6}) = \frac{\frac{6 x}{7 x - 6}}{7 (\frac{x}{7 x - 6}) - 1}$

Step 4.3.4

Simplify the denominator.

Tap for more steps...

Step 4.3.4.1

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

$f (\frac{x}{7 x - 6}) = \frac{\frac{6 x}{7 x - 6}}{\frac{7 x}{7 x - 6} - 1}$

Step 4.3.4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7 x - 6}{7 x - 6}$ .

$f (\frac{x}{7 x - 6}) = \frac{\frac{6 x}{7 x - 6}}{\frac{7 x}{7 x - 6} - 1 \cdot \frac{7 x - 6}{7 x - 6}}$

Step 4.3.4.3

Combine $- 1$ and $\frac{7 x - 6}{7 x - 6}$ .

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

Step 4.3.4.4

Combine the numerators over the common denominator.

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

Step 4.3.4.5

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

Tap for more steps...

Step 4.3.4.5.1

Apply the distributive property.

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

Step 4.3.4.5.2

Multiply $7$ by $- 1$ .

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

Step 4.3.4.5.3

Multiply $- 1$ by $- 6$ .

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

Step 4.3.4.5.4

Subtract $7 x$ from $7 x$ .

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

Step 4.3.4.5.5

Add $0$ and $6$ .

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

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.6

Cancel the common factor of $6$ .

Tap for more steps...

Step 4.3.6.1

Factor $6$ out of $6 x$ .

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

Step 4.3.6.2

Cancel the common factor.

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

Step 4.3.6.3

Rewrite the expression.

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

Step 4.3.7

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

Tap for more steps...

Step 4.3.7.1

Cancel the common factor.

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

Step 4.3.7.2

Rewrite the expression.

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

Step 4.4

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

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