Finite Math Examples

$x - 5 y = - 4$

Step 1

Subtract $x$ from both sides of the equation.

$- 5 y = - 4 - x$

Step 2

Divide each term in $- 5 y = - 4 - x$ by $- 5$ and simplify.

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Step 2.1

Divide each term in $- 5 y = - 4 - x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 4}{- 5} + \frac{- x}{- 5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 4}{- 5} + \frac{- x}{- 5}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4}{- 5} + \frac{- x}{- 5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{4}{5} + \frac{- x}{- 5}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{4}{5} + \frac{x}{5}$

Step 3

Interchange the variables.

$x = \frac{4}{5} + \frac{y}{5}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $\frac{4}{5} + \frac{y}{5} = x$ .

$\frac{4}{5} + \frac{y}{5} = x$

Step 4.2

Subtract $\frac{4}{5}$ from both sides of the equation.

$\frac{y}{5} = x - \frac{4}{5}$

Step 4.3

Multiply both sides of the equation by $5$ .

$5 \frac{y}{5} = 5 (x - \frac{4}{5})$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Cancel the common factor of $5$ .

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Step 4.4.1.1.1

Cancel the common factor.

$5 \frac{y}{5} = 5 (x - \frac{4}{5})$

Step 4.4.1.1.2

Rewrite the expression.

$y = 5 (x - \frac{4}{5})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $5 (x - \frac{4}{5})$ .

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Step 4.4.2.1.1

Apply the distributive property.

$y = 5 x + 5 (- \frac{4}{5})$

Step 4.4.2.1.2

Cancel the common factor of $5$ .

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Step 4.4.2.1.2.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$y = 5 x + 5 (\frac{- 4}{5})$

Step 4.4.2.1.2.2

Cancel the common factor.

$y = 5 x + 5 (\frac{- 4}{5})$

Step 4.4.2.1.2.3

Rewrite the expression.

$y = 5 x - 4$

Step 5

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

$f^{- 1} (x) = 5 x - 4$

Step 6

Verify if $f^{- 1} (x) = 5 x - 4$ is the inverse of $f (x) = \frac{4}{5} + \frac{x}{5}$ .

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Step 6.1

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

Step 6.2

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

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Step 6.2.1

Set up the composite result function.

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

Step 6.2.2

Evaluate $f^{- 1} (\frac{4}{5} + \frac{x}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = 5 (\frac{4}{5} + \frac{x}{5}) - 4$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Apply the distributive property.

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = 5 (\frac{4}{5}) + 5 (\frac{x}{5}) - 4$

Step 6.2.3.2

Cancel the common factor of $5$ .

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Step 6.2.3.2.1

Cancel the common factor.

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = 5 (\frac{4}{5}) + 5 (\frac{x}{5}) - 4$

Step 6.2.3.2.2

Rewrite the expression.

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = 4 + 5 (\frac{x}{5}) - 4$

Step 6.2.3.3

Cancel the common factor of $5$ .

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Step 6.2.3.3.1

Cancel the common factor.

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = 4 + 5 (\frac{x}{5}) - 4$

Step 6.2.3.3.2

Rewrite the expression.

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = 4 + x - 4$

Step 6.2.4

Combine the opposite terms in $4 + x - 4$ .

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Step 6.2.4.1

Subtract $4$ from $4$ .

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = x + 0$

Step 6.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{4}{5} + \frac{x}{5}) = x$

Step 6.3

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

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Step 6.3.1

Set up the composite result function.

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

Step 6.3.2

Evaluate $f (5 x - 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (5 x - 4) = \frac{4}{5} + \frac{5 x - 4}{5}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (5 x - 4) = \frac{4 + 5 x - 4}{5}$

Step 6.3.4

Combine the opposite terms in $4 + 5 x - 4$ .

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Step 6.3.4.1

Subtract $4$ from $4$ .

$f (5 x - 4) = \frac{5 x + 0}{5}$

Step 6.3.4.2

Add $5 x$ and $0$ .

$f (5 x - 4) = \frac{5 x}{5}$

Step 6.3.5

Cancel the common factor of $5$ .

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Step 6.3.5.1

Cancel the common factor.

$f (5 x - 4) = \frac{5 x}{5}$

Step 6.3.5.2

Divide $x$ by $1$ .

$f (5 x - 4) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 5 x - 4$ is the inverse of $f (x) = \frac{4}{5} + \frac{x}{5}$ .

$f^{- 1} (x) = 5 x - 4$