Algebra Examples

$C (f) = \frac{5}{9} (f - 32)$

Step 1

Write $C (f) = \frac{5}{9} (f - 32)$ as an equation.

$y = \frac{5}{9} (f - 32)$

Step 2

Interchange the variables.

$f = \frac{5}{9} \cdot (y - 32)$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{5}{9} \cdot (y - 32) = f$ .

$\frac{5}{9} \cdot (y - 32) = f$

Step 3.2

Multiply both sides of the equation by $\frac{9}{5}$ .

$\frac{9}{5} (\frac{5}{9} \cdot (y - 32)) = \frac{9}{5} f$

Step 3.3

Simplify both sides of the equation.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

Simplify $\frac{9}{5} (\frac{5}{9} \cdot (y - 32))$ .

Tap for more steps...

Step 3.3.1.1.1

Apply the distributive property.

$\frac{9}{5} (\frac{5}{9} y + \frac{5}{9} \cdot - 32) = \frac{9}{5} f$

Step 3.3.1.1.2

Combine $\frac{5}{9}$ and $y$ .

$\frac{9}{5} (\frac{5 y}{9} + \frac{5}{9} \cdot - 32) = \frac{9}{5} f$

Step 3.3.1.1.3

Multiply $\frac{5}{9} \cdot - 32$ .

Tap for more steps...

Step 3.3.1.1.3.1

Combine $\frac{5}{9}$ and $- 32$ .

$\frac{9}{5} (\frac{5 y}{9} + \frac{5 \cdot - 32}{9}) = \frac{9}{5} f$

Step 3.3.1.1.3.2

Multiply $5$ by $- 32$ .

$\frac{9}{5} (\frac{5 y}{9} + \frac{- 160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.4

Move the negative in front of the fraction.

$\frac{9}{5} (\frac{5 y}{9} - \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.5

Apply the distributive property.

$\frac{9}{5} \cdot \frac{5 y}{9} + \frac{9}{5} (- \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.6

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.3.1.1.6.1

Cancel the common factor.

$\frac{9}{5} \cdot \frac{5 y}{9} + \frac{9}{5} (- \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.6.2

Rewrite the expression.

$\frac{1}{5} (5 y) + \frac{9}{5} (- \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.7

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.1.1.7.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) + \frac{9}{5} (- \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.7.2

Cancel the common factor.

$\frac{1}{5} (5 y) + \frac{9}{5} (- \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.7.3

Rewrite the expression.

$y + \frac{9}{5} (- \frac{160}{9}) = \frac{9}{5} f$

Step 3.3.1.1.8

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.3.1.1.8.1

Move the leading negative in $- \frac{160}{9}$ into the numerator.

$y + \frac{9}{5} \cdot \frac{- 160}{9} = \frac{9}{5} f$

Step 3.3.1.1.8.2

Cancel the common factor.

$y + \frac{9}{5} \cdot \frac{- 160}{9} = \frac{9}{5} f$

Step 3.3.1.1.8.3

Rewrite the expression.

$y + \frac{1}{5} \cdot - 160 = \frac{9}{5} f$

Step 3.3.1.1.9

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.1.1.9.1

Factor $5$ out of $- 160$ .

$y + \frac{1}{5} \cdot (5 (- 32)) = \frac{9}{5} f$

Step 3.3.1.1.9.2

Cancel the common factor.

$y + \frac{1}{5} \cdot (5 \cdot - 32) = \frac{9}{5} f$

Step 3.3.1.1.9.3

Rewrite the expression.

$y - 32 = \frac{9}{5} f$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Combine $\frac{9}{5}$ and $f$ .

$y - 32 = \frac{9 f}{5}$

Step 3.4

Add $32$ to both sides of the equation.

$y = \frac{9 f}{5} + 32$

Step 4

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

$C^{- 1} (f) = \frac{9 f}{5} + 32$

Step 5

Verify if $C^{- 1} (f) = \frac{9 f}{5} + 32$ is the inverse of $C (f) = \frac{5}{9} \cdot (f - 32)$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $C^{- 1} (\frac{5}{9} \cdot (f - 32))$ by substituting in the value of $C$ into $C^{- 1}$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = \frac{9 (\frac{5}{9} \cdot (f - 32))}{5} + 32$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Combine $9$ and $\frac{5}{9}$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = \frac{\frac{9 \cdot 5}{9} \cdot (f - 32)}{5} + 32$

Step 5.2.3.2

Multiply $9$ by $5$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = \frac{\frac{45}{9} \cdot (f - 32)}{5} + 32$

Step 5.2.3.3

Divide $45$ by $9$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = \frac{5 (f - 32)}{5} + 32$

Step 5.2.3.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.4.1

Cancel the common factor.

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = \frac{5 (f - 32)}{5} + 32$

Step 5.2.3.4.2

Divide $f - 32$ by $1$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = f - 32 + 32$

Step 5.2.4

Combine the opposite terms in $f - 32 + 32$ .

Tap for more steps...

Step 5.2.4.1

Add $- 32$ and $32$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = f + 0$

Step 5.2.4.2

Add $f$ and $0$ .

$C^{- 1} (\frac{5}{9} \cdot (f - 32)) = f$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $C (\frac{9 f}{5} + 32)$ by substituting in the value of $C^{- 1}$ into $C$ .

$C (\frac{9 f}{5} + 32) = \frac{5}{9} \cdot ((\frac{9 f}{5} + 32) - 32)$

Step 5.3.3

Combine the opposite terms in $(\frac{9 f}{5} + 32) - 32$ .

Tap for more steps...

Step 5.3.3.1

Subtract $32$ from $32$ .

$C (\frac{9 f}{5} + 32) = \frac{5}{9} \cdot (\frac{9 f}{5} + 0)$

Step 5.3.3.2

Add $\frac{9 f}{5}$ and $0$ .

$C (\frac{9 f}{5} + 32) = \frac{5}{9} \cdot \frac{9 f}{5}$

Step 5.3.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.4.1

Cancel the common factor.

$C (\frac{9 f}{5} + 32) = \frac{5}{9} \cdot \frac{9 f}{5}$

Step 5.3.4.2

Rewrite the expression.

$C (\frac{9 f}{5} + 32) = \frac{1}{9} \cdot (9 f)$

Step 5.3.5

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.3.5.1

Factor $9$ out of $9 f$ .

$C (\frac{9 f}{5} + 32) = \frac{1}{9} \cdot (9 (f))$

Step 5.3.5.2

Cancel the common factor.

$C (\frac{9 f}{5} + 32) = \frac{1}{9} \cdot (9 f)$

Step 5.3.5.3

Rewrite the expression.

$C (\frac{9 f}{5} + 32) = f$

Step 5.4

Since $C^{- 1} (C (f)) = f$ and $C (C^{- 1} (f)) = f$ , then $C^{- 1} (f) = \frac{9 f}{5} + 32$ is the inverse of $C (f) = \frac{5}{9} \cdot (f - 32)$ .

$C^{- 1} (f) = \frac{9 f}{5} + 32$