Algebra Examples

$y = \frac{x + 5}{7}$

Step 1

Interchange the variables.

$x = \frac{y + 5}{7}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

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

$\frac{y + 5}{7} = x$

Step 2.2

Multiply both sides of the equation by $7$ .

$7 \frac{y + 5}{7} = 7 x$

Step 2.3

Simplify the left side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.3.1.1

Cancel the common factor.

$7 \frac{y + 5}{7} = 7 x$

Step 2.3.1.2

Rewrite the expression.

$y + 5 = 7 x$

Step 2.4

Subtract $5$ from both sides of the equation.

$y = 7 x - 5$

Step 3

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

$f^{- 1} (x) = 7 x - 5$

Step 4

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

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

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

Step 4.2.3

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Rewrite the expression.

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

Step 4.2.4

Combine the opposite terms in $x + 5 - 5$ .

Tap for more steps...

Step 4.2.4.1

Subtract $5$ from $5$ .

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

Step 4.2.4.2

Add $x$ and $0$ .

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

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 $f (7 x - 5)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Simplify the numerator.

Tap for more steps...

Step 4.3.3.1

Add $- 5$ and $5$ .

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

Step 4.3.3.2

Add $7 x$ and $0$ .

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

Step 4.3.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Divide $x$ by $1$ .

$f (7 x - 5) = x$

Step 4.4

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

$f^{- 1} (x) = 7 x - 5$