Finite Math Examples

$4 x - 3 y = 24$

Step 1

Subtract $4 x$ from both sides of the equation.

$- 3 y = 24 - 4 x$

Step 2

Divide each term in $- 3 y = 24 - 4 x$ by $- 3$ and simplify.

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Step 2.1

Divide each term in $- 3 y = 24 - 4 x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{24}{- 3} + \frac{- 4 x}{- 3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{24}{- 3} + \frac{- 4 x}{- 3}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{24}{- 3} + \frac{- 4 x}{- 3}$

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

Divide $24$ by $- 3$ .

$y = - 8 + \frac{- 4 x}{- 3}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y = - 8 + \frac{4 x}{3}$

Step 3

Interchange the variables.

$x = - 8 + \frac{4 y}{3}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $- 8 + \frac{4 y}{3} = x$ .

$- 8 + \frac{4 y}{3} = x$

Step 4.2

Add $8$ to both sides of the equation.

$\frac{4 y}{3} = x + 8$

Step 4.3

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x + 8)$

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

Simplify $\frac{3}{4} \cdot \frac{4 y}{3}$ .

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Step 4.4.1.1.1

Cancel the common factor of $3$ .

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Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x + 8)$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 y) = \frac{3}{4} (x + 8)$

Step 4.4.1.1.2

Cancel the common factor of $4$ .

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Step 4.4.1.1.2.1

Factor $4$ out of $4 y$ .

$\frac{1}{4} (4 (y)) = \frac{3}{4} (x + 8)$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 y) = \frac{3}{4} (x + 8)$

Step 4.4.1.1.2.3

Rewrite the expression.

$y = \frac{3}{4} (x + 8)$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $\frac{3}{4} (x + 8)$ .

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Step 4.4.2.1.1

Apply the distributive property.

$y = \frac{3}{4} x + \frac{3}{4} \cdot 8$

Step 4.4.2.1.2

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

$y = \frac{3 x}{4} + \frac{3}{4} \cdot 8$

Step 4.4.2.1.3

Cancel the common factor of $4$ .

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Step 4.4.2.1.3.1

Factor $4$ out of $8$ .

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

Step 4.4.2.1.3.2

Cancel the common factor.

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

Step 4.4.2.1.3.3

Rewrite the expression.

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

Step 4.4.2.1.4

Multiply $3$ by $2$ .

$y = \frac{3 x}{4} + 6$

Step 5

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

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

Step 6

Verify if $f^{- 1} (x) = \frac{3 x}{4} + 6$ is the inverse of $f (x) = - 8 + \frac{4 x}{3}$ .

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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 $f^{- 1} (- 8 + \frac{4 x}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = \frac{3 (- 8 + \frac{4 x}{3})}{4} + 6$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Cancel the common factor of $- 8 + \frac{4 x}{3}$ and $4$ .

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Step 6.2.3.1.1

Factor $4$ out of $3 (- 8 + \frac{4 x}{3})$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = \frac{4 (3 (- 2 + \frac{x}{3}))}{4} + 6$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $4$ out of $4$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = \frac{4 (3 (- 2 + \frac{x}{3}))}{4 (1)} + 6$

Step 6.2.3.1.2.2

Cancel the common factor.

$f^{- 1} (- 8 + \frac{4 x}{3}) = \frac{4 (3 (- 2 + \frac{x}{3}))}{4 \cdot 1} + 6$

Step 6.2.3.1.2.3

Rewrite the expression.

$f^{- 1} (- 8 + \frac{4 x}{3}) = \frac{3 (- 2 + \frac{x}{3})}{1} + 6$

Step 6.2.3.1.2.4

Divide $3 (- 2 + \frac{x}{3})$ by $1$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = 3 (- 2 + \frac{x}{3}) + 6$

Step 6.2.3.2

Apply the distributive property.

$f^{- 1} (- 8 + \frac{4 x}{3}) = 3 \cdot - 2 + 3 (\frac{x}{3}) + 6$

Step 6.2.3.3

Multiply $3$ by $- 2$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = - 6 + 3 (\frac{x}{3}) + 6$

Step 6.2.3.4

Cancel the common factor of $3$ .

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Step 6.2.3.4.1

Cancel the common factor.

$f^{- 1} (- 8 + \frac{4 x}{3}) = - 6 + 3 (\frac{x}{3}) + 6$

Step 6.2.3.4.2

Rewrite the expression.

$f^{- 1} (- 8 + \frac{4 x}{3}) = - 6 + x + 6$

Step 6.2.4

Combine the opposite terms in $- 6 + x + 6$ .

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Step 6.2.4.1

Add $- 6$ and $6$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = x + 0$

Step 6.2.4.2

Add $x$ and $0$ .

$f^{- 1} (- 8 + \frac{4 x}{3}) = x$

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 $f (\frac{3 x}{4} + 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{4} + 6) = - 8 + \frac{4 (\frac{3 x}{4} + 6)}{3}$

Step 6.3.3

Simplify each term.

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Step 6.3.3.1

Cancel the common factor of $\frac{3 x}{4} + 6$ and $3$ .

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Step 6.3.3.1.1

Factor $3$ out of $4 (\frac{3 x}{4} + 6)$ .

$f (\frac{3 x}{4} + 6) = - 8 + \frac{3 (4 (\frac{x}{4} + 2))}{3}$

Step 6.3.3.1.2

Cancel the common factors.

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Step 6.3.3.1.2.1

Factor $3$ out of $3$ .

$f (\frac{3 x}{4} + 6) = - 8 + \frac{3 (4 (\frac{x}{4} + 2))}{3 (1)}$

Step 6.3.3.1.2.2

Cancel the common factor.

$f (\frac{3 x}{4} + 6) = - 8 + \frac{3 (4 (\frac{x}{4} + 2))}{3 \cdot 1}$

Step 6.3.3.1.2.3

Rewrite the expression.

$f (\frac{3 x}{4} + 6) = - 8 + \frac{4 (\frac{x}{4} + 2)}{1}$

Step 6.3.3.1.2.4

Divide $4 (\frac{x}{4} + 2)$ by $1$ .

$f (\frac{3 x}{4} + 6) = - 8 + 4 (\frac{x}{4} + 2)$

Step 6.3.3.2

Apply the distributive property.

$f (\frac{3 x}{4} + 6) = - 8 + 4 (\frac{x}{4}) + 4 \cdot 2$

Step 6.3.3.3

Cancel the common factor of $4$ .

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Step 6.3.3.3.1

Cancel the common factor.

$f (\frac{3 x}{4} + 6) = - 8 + 4 (\frac{x}{4}) + 4 \cdot 2$

Step 6.3.3.3.2

Rewrite the expression.

$f (\frac{3 x}{4} + 6) = - 8 + x + 4 \cdot 2$

Step 6.3.3.4

Multiply $4$ by $2$ .

$f (\frac{3 x}{4} + 6) = - 8 + x + 8$

Step 6.3.4

Combine the opposite terms in $- 8 + x + 8$ .

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Step 6.3.4.1

Add $- 8$ and $8$ .

$f (\frac{3 x}{4} + 6) = x + 0$

Step 6.3.4.2

Add $x$ and $0$ .

$f (\frac{3 x}{4} + 6) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{4} + 6$ is the inverse of $f (x) = - 8 + \frac{4 x}{3}$ .

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