Algebra Examples

$y = \frac{9 x}{10}$

Step 1

Interchange the variables.

$x = \frac{9 y}{10}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{9 y}{10} = x$ .

$\frac{9 y}{10} = x$

Step 2.2

Multiply both sides of the equation by $\frac{10}{9}$ .

$\frac{10}{9} \cdot \frac{9 y}{10} = \frac{10}{9} x$

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{10}{9} \cdot \frac{9 y}{10}$ .

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Step 2.3.1.1.1

Cancel the common factor of $10$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

$\frac{10}{9} \cdot \frac{9 y}{10} = \frac{10}{9} x$

Step 2.3.1.1.1.2

Rewrite the expression.

$\frac{1}{9} (9 y) = \frac{10}{9} x$

Step 2.3.1.1.2

Cancel the common factor of $9$ .

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Step 2.3.1.1.2.1

Factor $9$ out of $9 y$ .

$\frac{1}{9} (9 (y)) = \frac{10}{9} x$

Step 2.3.1.1.2.2

Cancel the common factor.

$\frac{1}{9} (9 y) = \frac{10}{9} x$

Step 2.3.1.1.2.3

Rewrite the expression.

$y = \frac{10}{9} x$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Combine $\frac{10}{9}$ and $x$ .

$y = \frac{10 x}{9}$

Step 3

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

$f^{- 1} (x) = \frac{10 x}{9}$

Step 4

Verify if $f^{- 1} (x) = \frac{10 x}{9}$ is the inverse of $f (x) = \frac{9 x}{10}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{9 x}{10})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{9 x}{10}) = \frac{10 (\frac{9 x}{10})}{9}$

Step 4.2.3

Combine $10$ and $\frac{9 x}{10}$ .

$f^{- 1} (\frac{9 x}{10}) = \frac{\frac{10 (9 x)}{10}}{9}$

Step 4.2.4

Multiply $10$ by $9$ .

$f^{- 1} (\frac{9 x}{10}) = \frac{\frac{90 x}{10}}{9}$

Step 4.2.5

Reduce the expression by cancelling the common factors.

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Step 4.2.5.1

Reduce the expression $\frac{90 x}{10}$ by cancelling the common factors.

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Step 4.2.5.1.1

Factor $10$ out of $90 x$ .

$f^{- 1} (\frac{9 x}{10}) = \frac{\frac{10 (9 x)}{10}}{9}$

Step 4.2.5.1.2

Factor $10$ out of $10$ .

$f^{- 1} (\frac{9 x}{10}) = \frac{\frac{10 (9 x)}{10 (1)}}{9}$

Step 4.2.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{9 x}{10}) = \frac{\frac{10 (9 x)}{10 \cdot 1}}{9}$

Step 4.2.5.1.4

Rewrite the expression.

$f^{- 1} (\frac{9 x}{10}) = \frac{\frac{9 x}{1}}{9}$

Step 4.2.5.2

Divide $9 x$ by $1$ .

$f^{- 1} (\frac{9 x}{10}) = \frac{9 x}{9}$

Step 4.2.6

Cancel the common factor of $9$ .

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Step 4.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{9 x}{10}) = \frac{9 x}{9}$

Step 4.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{9 x}{10}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{10 x}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{10 x}{9}) = \frac{9 (\frac{10 x}{9})}{10}$

Step 4.3.3

Combine $9$ and $\frac{10 x}{9}$ .

$f (\frac{10 x}{9}) = \frac{\frac{9 (10 x)}{9}}{10}$

Step 4.3.4

Multiply $9$ by $10$ .

$f (\frac{10 x}{9}) = \frac{\frac{90 x}{9}}{10}$

Step 4.3.5

Reduce the expression by cancelling the common factors.

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Step 4.3.5.1

Reduce the expression $\frac{90 x}{9}$ by cancelling the common factors.

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Step 4.3.5.1.1

Factor $9$ out of $90 x$ .

$f (\frac{10 x}{9}) = \frac{\frac{9 (10 x)}{9}}{10}$

Step 4.3.5.1.2

Factor $9$ out of $9$ .

$f (\frac{10 x}{9}) = \frac{\frac{9 (10 x)}{9 (1)}}{10}$

Step 4.3.5.1.3

Cancel the common factor.

$f (\frac{10 x}{9}) = \frac{\frac{9 (10 x)}{9 \cdot 1}}{10}$

Step 4.3.5.1.4

Rewrite the expression.

$f (\frac{10 x}{9}) = \frac{\frac{10 x}{1}}{10}$

Step 4.3.5.2

Divide $10 x$ by $1$ .

$f (\frac{10 x}{9}) = \frac{10 x}{10}$

Step 4.3.6

Cancel the common factor of $10$ .

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Step 4.3.6.1

Cancel the common factor.

$f (\frac{10 x}{9}) = \frac{10 x}{10}$

Step 4.3.6.2

Divide $x$ by $1$ .

$f (\frac{10 x}{9}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{10 x}{9}$ is the inverse of $f (x) = \frac{9 x}{10}$ .

$f^{- 1} (x) = \frac{10 x}{9}$