Algebra Examples

$y = \frac{1}{2} - \frac{2}{3} x$

Step 1

Interchange the variables.

$x = \frac{1}{2} - \frac{2}{3} y$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{2} - \frac{2}{3} y = x$ .

$\frac{1}{2} - \frac{2}{3} y = x$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $y$ and $\frac{2}{3}$ .

$\frac{1}{2} - \frac{y \cdot 2}{3} = x$

Step 2.2.2

Move $2$ to the left of $y$ .

$\frac{1}{2} - \frac{2 y}{3} = x$

Step 2.3

Subtract $\frac{1}{2}$ from both sides of the equation.

$- \frac{2 y}{3} = x - \frac{1}{2}$

Step 2.4

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 y}{3}) = - \frac{3}{2} (x - \frac{1}{2})$

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 $- \frac{3}{2} (- \frac{2 y}{3})$ .

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Step 2.5.1.1.1

Cancel the common factor of $3$ .

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Step 2.5.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 y}{3}) = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.1.2

Move the leading negative in $- \frac{2 y}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 y) = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.1.1.2.1

Factor $2$ out of $- 2 y$ .

$\frac{- 1}{2} (2 (- y)) = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- y)) = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.3

Multiply.

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Step 2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{3}{2} (x - \frac{1}{2})$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $- \frac{3}{2} (x - \frac{1}{2})$ .

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Step 2.5.2.1.1

Apply the distributive property.

$y = - \frac{3}{2} x - \frac{3}{2} (- \frac{1}{2})$

Step 2.5.2.1.2

Combine $x$ and $\frac{3}{2}$ .

$y = - \frac{x \cdot 3}{2} - \frac{3}{2} (- \frac{1}{2})$

Step 2.5.2.1.3

Multiply $- \frac{3}{2} (- \frac{1}{2})$ .

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Step 2.5.2.1.3.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{x \cdot 3}{2} + 1 (\frac{3}{2}) \frac{1}{2}$

Step 2.5.2.1.3.2

Multiply $\frac{3}{2}$ by $1$ .

$y = - \frac{x \cdot 3}{2} + \frac{3}{2} \cdot \frac{1}{2}$

Step 2.5.2.1.3.3

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$y = - \frac{x \cdot 3}{2} + \frac{3}{2 \cdot 2}$

Step 2.5.2.1.3.4

Multiply $2$ by $2$ .

$y = - \frac{x \cdot 3}{2} + \frac{3}{4}$

Step 2.5.2.1.4

Move $3$ to the left of $x$ .

$y = - \frac{3 x}{2} + \frac{3}{4}$

Step 3

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

$f^{- 1} (x) = - \frac{3 x}{2} + \frac{3}{4}$

Step 4

Verify if $f^{- 1} (x) = - \frac{3 x}{2} + \frac{3}{4}$ is the inverse of $f (x) = \frac{1}{2} - \frac{2}{3} x$ .

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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 $f^{- 1} (\frac{1}{2} - \frac{2}{3} x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{1}{2} - \frac{2}{3} x)}{2} + \frac{3}{4}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Combine $x$ and $\frac{2}{3}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{1}{2} - \frac{x \cdot 2}{3})}{2} + \frac{3}{4}$

Step 4.2.3.1.2

Move $2$ to the left of $x$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{1}{2} - \frac{2 \cdot x}{3})}{2} + \frac{3}{4}$

Step 4.2.3.1.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{1}{2} \cdot \frac{3}{3} - \frac{2 x}{3})}{2} + \frac{3}{4}$

Step 4.2.3.1.4

To write $- \frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{1}{2} \cdot \frac{3}{3} - \frac{2 x}{3} \cdot \frac{2}{2})}{2} + \frac{3}{4}$

Step 4.2.3.1.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1.5.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{3}{2 \cdot 3} - \frac{2 x}{3} \cdot \frac{2}{2})}{2} + \frac{3}{4}$

Step 4.2.3.1.5.2

Multiply $2$ by $3$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{3}{6} - \frac{2 x}{3} \cdot \frac{2}{2})}{2} + \frac{3}{4}$

Step 4.2.3.1.5.3

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{3}{6} - \frac{2 x \cdot 2}{3 \cdot 2})}{2} + \frac{3}{4}$

Step 4.2.3.1.5.4

Multiply $3$ by $2$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{3}{6} - \frac{2 x \cdot 2}{6})}{2} + \frac{3}{4}$

Step 4.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{3 - 2 x \cdot 2}{6})}{2} + \frac{3}{4}$

Step 4.2.3.1.7

Multiply $2$ by $- 2$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 (\frac{3 - 4 x}{6})}{2} + \frac{3}{4}$

Step 4.2.3.2

Combine $3$ and $\frac{3 - 4 x}{6}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{\frac{3 (3 - 4 x)}{6}}{2} + \frac{3}{4}$

Step 4.2.3.3

Reduce the expression $\frac{3 (3 - 4 x)}{6}$ by cancelling the common factors.

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Step 4.2.3.3.1

Factor $3$ out of $6$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{\frac{3 (3 - 4 x)}{3 \cdot 2}}{2} + \frac{3}{4}$

Step 4.2.3.3.2

Cancel the common factor.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{\frac{3 (3 - 4 x)}{3 \cdot 2}}{2} + \frac{3}{4}$

Step 4.2.3.3.3

Rewrite the expression.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{\frac{3 - 4 x}{2}}{2} + \frac{3}{4}$

Step 4.2.3.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - (\frac{3 - 4 x}{2} \cdot \frac{1}{2}) + \frac{3}{4}$

Step 4.2.3.5

Multiply $\frac{3 - 4 x}{2} \cdot \frac{1}{2}$ .

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Step 4.2.3.5.1

Multiply $\frac{3 - 4 x}{2}$ by $\frac{1}{2}$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 - 4 x}{2 \cdot 2} + \frac{3}{4}$

Step 4.2.3.5.2

Multiply $2$ by $2$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = - \frac{3 - 4 x}{4} + \frac{3}{4}$

Step 4.2.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{- (3 - 4 x) + 3}{4}$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Apply the distributive property.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{- 1 \cdot 3 - (- 4 x) + 3}{4}$

Step 4.2.5.2

Multiply $- 1$ by $3$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{- 3 - (- 4 x) + 3}{4}$

Step 4.2.5.3

Multiply $- 4$ by $- 1$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{- 3 + 4 x + 3}{4}$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine the opposite terms in $- 3 + 4 x + 3$ .

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Step 4.2.6.1.1

Add $- 3$ and $3$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{4 x + 0}{4}$

Step 4.2.6.1.2

Add $4 x$ and $0$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{4 x}{4}$

Step 4.2.6.2

Cancel the common factor of $4$ .

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Step 4.2.6.2.1

Cancel the common factor.

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = \frac{4 x}{4}$

Step 4.2.6.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{1}{2} - \frac{2}{3} x) = x$

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 $f (- \frac{3 x}{2} + \frac{3}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} - \frac{2}{3} \cdot (- \frac{3 x}{2} + \frac{3}{4})$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} - \frac{2}{3} \cdot (- \frac{3 x}{2}) - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.2

Cancel the common factor of $2$ .

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Step 4.3.3.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{- 2}{3} \cdot (- \frac{3 x}{2}) - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.2.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{- 2}{3} \cdot \frac{- 3 x}{2} - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.2.3

Factor $2$ out of $- 2$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{2 (- 1)}{3} \cdot \frac{- 3 x}{2} - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.2.4

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{2 \cdot - 1}{3} \cdot \frac{- 3 x}{2} - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.2.5

Rewrite the expression.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{- 1}{3} \cdot (- 3 x) - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.3

Cancel the common factor of $3$ .

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Step 4.3.3.3.1

Factor $3$ out of $- 3 x$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{- 1}{3} \cdot (3 (- x)) - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.3.2

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + \frac{- 1}{3} \cdot (3 (- x)) - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.3.3

Rewrite the expression.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} - 1 (- x) - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.4

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + 1 x - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.5

Multiply $x$ by $1$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x - \frac{2}{3} \cdot \frac{3}{4}$

Step 4.3.3.6

Cancel the common factor of $2$ .

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Step 4.3.3.6.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x + \frac{- 2}{3} \cdot \frac{3}{4}$

Step 4.3.3.6.2

Factor $2$ out of $- 2$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x + \frac{2 (- 1)}{3} \cdot \frac{3}{4}$

Step 4.3.3.6.3

Factor $2$ out of $4$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x + \frac{2 \cdot - 1}{3} \cdot \frac{3}{2 \cdot 2}$

Step 4.3.3.6.4

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x + \frac{2 \cdot - 1}{3} \cdot \frac{3}{2 \cdot 2}$

Step 4.3.3.6.5

Rewrite the expression.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x + \frac{- 1}{3} \cdot \frac{3}{2}$

Step 4.3.3.7

Cancel the common factor of $3$ .

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Step 4.3.3.7.1

Cancel the common factor.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x + \frac{- 1}{3} \cdot \frac{3}{2}$

Step 4.3.3.7.2

Rewrite the expression.

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x - 1 (\frac{1}{2})$

Step 4.3.3.8

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = \frac{1}{2} + x - \frac{1}{2}$

Step 4.3.4

Combine the opposite terms in $\frac{1}{2} + x - \frac{1}{2}$ .

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Step 4.3.4.1

Combine the numerators over the common denominator.

$f (- \frac{3 x}{2} + \frac{3}{4}) = x + \frac{1 - 1}{2}$

Step 4.3.4.2

Subtract $1$ from $1$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = x + \frac{0}{2}$

Step 4.3.4.3

Divide $0$ by $2$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = x + 0$

Step 4.3.4.4

Add $x$ and $0$ .

$f (- \frac{3 x}{2} + \frac{3}{4}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{3 x}{2} + \frac{3}{4}$ is the inverse of $f (x) = \frac{1}{2} - \frac{2}{3} x$ .

$f^{- 1} (x) = - \frac{3 x}{2} + \frac{3}{4}$