Algebra Examples

$y = - \frac{4}{5} x + \frac{1}{5}$

Step 1

Interchange the variables.

$x = - \frac{4}{5} y + \frac{1}{5}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \frac{4}{5} y + \frac{1}{5} = x$ .

$- \frac{4}{5} y + \frac{1}{5} = x$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $y$ and $\frac{4}{5}$ .

$- \frac{y \cdot 4}{5} + \frac{1}{5} = x$

Step 2.2.2

Move $4$ to the left of $y$ .

$- \frac{4 y}{5} + \frac{1}{5} = x$

Step 2.3

Subtract $\frac{1}{5}$ from both sides of the equation.

$- \frac{4 y}{5} = x - \frac{1}{5}$

Step 2.4

Multiply both sides of the equation by $- \frac{5}{4}$ .

$- \frac{5}{4} (- \frac{4 y}{5}) = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

Cancel the common factor of $5$ .

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Step 2.5.1.1.1.1

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

$\frac{- 5}{4} (- \frac{4 y}{5}) = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.1.2

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

$\frac{- 5}{4} \cdot \frac{- 4 y}{5} = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{4} \cdot \frac{- 4 y}{5} = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{4} \cdot \frac{- 4 y}{5} = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 y) = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.2

Cancel the common factor of $4$ .

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Step 2.5.1.1.2.1

Factor $4$ out of $- 4 y$ .

$\frac{- 1}{4} (4 (- y)) = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- y)) = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.3

Multiply.

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Step 2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{5}{4} (x - \frac{1}{5})$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $- \frac{5}{4} (x - \frac{1}{5})$ .

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Step 2.5.2.1.1

Simplify terms.

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Step 2.5.2.1.1.1

Apply the distributive property.

$y = - \frac{5}{4} x - \frac{5}{4} (- \frac{1}{5})$

Step 2.5.2.1.1.2

Combine $x$ and $\frac{5}{4}$ .

$y = - \frac{x \cdot 5}{4} - \frac{5}{4} (- \frac{1}{5})$

Step 2.5.2.1.1.3

Cancel the common factor of $5$ .

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Step 2.5.2.1.1.3.1

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

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{4} (- \frac{1}{5})$

Step 2.5.2.1.1.3.2

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{4} \cdot \frac{- 1}{5}$

Step 2.5.2.1.1.3.3

Factor $5$ out of $- 5$ .

$y = - \frac{x \cdot 5}{4} + \frac{5 (- 1)}{4} \cdot \frac{- 1}{5}$

Step 2.5.2.1.1.3.4

Cancel the common factor.

$y = - \frac{x \cdot 5}{4} + \frac{5 \cdot - 1}{4} \cdot \frac{- 1}{5}$

Step 2.5.2.1.1.3.5

Rewrite the expression.

$y = - \frac{x \cdot 5}{4} + \frac{- 1}{4} \cdot - 1$

Step 2.5.2.1.1.4

Combine $\frac{- 1}{4}$ and $- 1$ .

$y = - \frac{x \cdot 5}{4} + \frac{- - 1}{4}$

Step 2.5.2.1.1.5

Multiply $- 1$ by $- 1$ .

$y = - \frac{x \cdot 5}{4} + \frac{1}{4}$

Step 2.5.2.1.2

Move $5$ to the left of $x$ .

$y = - \frac{5 x}{4} + \frac{1}{4}$

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Simplify each term.

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Step 4.2.4.1.1

Combine $x$ and $\frac{4}{5}$ .

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

Step 4.2.4.1.2

Move $4$ to the left of $x$ .

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

Step 4.2.4.2

Apply the distributive property.

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

Step 4.2.4.3

Cancel the common factor of $5$ .

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Step 4.2.4.3.1

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

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

Step 4.2.4.3.2

Factor $5$ out of $- 5$ .

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

Step 4.2.4.3.3

Cancel the common factor.

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

Step 4.2.4.3.4

Rewrite the expression.

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

Step 4.2.4.4

Multiply $- 4$ by $- 1$ .

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

Step 4.2.4.5

Cancel the common factor of $5$ .

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Step 4.2.4.5.1

Factor $5$ out of $- 5$ .

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

Step 4.2.4.5.2

Cancel the common factor.

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

Step 4.2.4.5.3

Rewrite the expression.

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

Step 4.2.5

Simplify terms.

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Step 4.2.5.1

Combine the opposite terms in $4 x - 1 + 1$ .

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Step 4.2.5.1.1

Add $- 1$ and $1$ .

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

Step 4.2.5.1.2

Add $4 x$ and $0$ .

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

Step 4.2.5.2

Cancel the common factor of $4$ .

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Step 4.2.5.2.1

Cancel the common factor.

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

Step 4.2.5.2.2

Divide $x$ by $1$ .

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

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Cancel the common factor of $4$ .

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

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

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

Step 4.3.3.2.3

Factor $4$ out of $- 4$ .

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

Step 4.3.3.2.4

Cancel the common factor.

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

Step 4.3.3.2.5

Rewrite the expression.

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

Step 4.3.3.3

Cancel the common factor of $5$ .

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Step 4.3.3.3.1

Factor $5$ out of $- 5 x$ .

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

Step 4.3.3.3.2

Cancel the common factor.

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

Step 4.3.3.3.3

Rewrite the expression.

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

Step 4.3.3.4

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.5

Multiply $x$ by $1$ .

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

Step 4.3.3.6

Cancel the common factor of $4$ .

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Step 4.3.3.6.1

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

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

Step 4.3.3.6.2

Factor $4$ out of $- 4$ .

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

Step 4.3.3.6.3

Cancel the common factor.

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

Step 4.3.3.6.4

Rewrite the expression.

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

Step 4.3.3.7

Move the negative in front of the fraction.

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

Step 4.3.4

Combine the opposite terms in $x - \frac{1}{5} + \frac{1}{5}$ .

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Step 4.3.4.1

Add $- \frac{1}{5}$ and $\frac{1}{5}$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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