Finite Math Examples

$4 x + 7 y$

Step 1

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

$7 y = - 4 x$

Step 2

Divide each term in $7 y = - 4 x$ by $7$ and simplify.

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Step 2.1

Divide each term in $7 y = - 4 x$ by $7$ .

$\frac{7 y}{7} = \frac{- 4 x}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{- 4 x}{7}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 x}{7}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y = - \frac{4 x}{7}$

Step 3

Interchange the variables.

$x = - \frac{4 y}{7}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $- \frac{4 y}{7} = x$ .

$- \frac{4 y}{7} = x$

Step 4.2

Multiply both sides of the equation by $- \frac{7}{4}$ .

$- \frac{7}{4} (- \frac{4 y}{7}) = - \frac{7}{4} x$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $- \frac{7}{4} (- \frac{4 y}{7})$ .

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Step 4.3.1.1.1

Cancel the common factor of $7$ .

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Step 4.3.1.1.1.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$\frac{- 7}{4} (- \frac{4 y}{7}) = - \frac{7}{4} x$

Step 4.3.1.1.1.2

Move the leading negative in $- \frac{4 y}{7}$ into the numerator.

$\frac{- 7}{4} \cdot \frac{- 4 y}{7} = - \frac{7}{4} x$

Step 4.3.1.1.1.3

Factor $7$ out of $- 7$ .

$\frac{7 (- 1)}{4} \cdot \frac{- 4 y}{7} = - \frac{7}{4} x$

Step 4.3.1.1.1.4

Cancel the common factor.

$\frac{7 \cdot - 1}{4} \cdot \frac{- 4 y}{7} = - \frac{7}{4} x$

Step 4.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 y) = - \frac{7}{4} x$

Step 4.3.1.1.2

Cancel the common factor of $4$ .

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Step 4.3.1.1.2.1

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

$\frac{- 1}{4} (4 (- y)) = - \frac{7}{4} x$

Step 4.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- y)) = - \frac{7}{4} x$

Step 4.3.1.1.2.3

Rewrite the expression.

$- - y = - \frac{7}{4} x$

Step 4.3.1.1.3

Multiply.

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Step 4.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{7}{4} x$

Step 4.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{7}{4} x$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $- \frac{7}{4} x$ .

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Step 4.3.2.1.1

Combine $x$ and $\frac{7}{4}$ .

$y = - \frac{x \cdot 7}{4}$

Step 4.3.2.1.2

Move $7$ to the left of $x$ .

$y = - \frac{7 x}{4}$

Step 5

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

$f^{- 1} (x) = - \frac{7 x}{4}$

Step 6

Verify if $f^{- 1} (x) = - \frac{7 x}{4}$ is the inverse of $f (x) = - \frac{4 x}{7}$ .

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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 $f^{- 1} (- \frac{4 x}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{7 (- \frac{4 x}{7})}{4}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

Multiply $- 1$ by $7$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{- 7 \frac{4 x}{7}}{4}$

Step 6.2.3.2

Combine $- 7$ and $\frac{4 x}{7}$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{\frac{- 7 (4 x)}{7}}{4}$

Step 6.2.4

Multiply $- 7$ by $4$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{\frac{- 28 x}{7}}{4}$

Step 6.2.5

Reduce the expression by cancelling the common factors.

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Step 6.2.5.1

Reduce the expression $\frac{- 28 x}{7}$ by cancelling the common factors.

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Step 6.2.5.1.1

Factor $7$ out of $- 28 x$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{\frac{7 (- 4 x)}{7}}{4}$

Step 6.2.5.1.2

Factor $7$ out of $7$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{\frac{7 (- 4 x)}{7 (1)}}{4}$

Step 6.2.5.1.3

Cancel the common factor.

$f^{- 1} (- \frac{4 x}{7}) = - \frac{\frac{7 (- 4 x)}{7 \cdot 1}}{4}$

Step 6.2.5.1.4

Rewrite the expression.

$f^{- 1} (- \frac{4 x}{7}) = - \frac{\frac{- 4 x}{1}}{4}$

Step 6.2.5.2

Divide $- 4 x$ by $1$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{- 4 x}{4}$

Step 6.2.6

Cancel the common factor of $- 4$ and $4$ .

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Step 6.2.6.1

Factor $4$ out of $- 4 x$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{4 (- x)}{4}$

Step 6.2.6.2

Cancel the common factors.

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Step 6.2.6.2.1

Factor $4$ out of $4$ .

$f^{- 1} (- \frac{4 x}{7}) = - \frac{4 (- x)}{4 (1)}$

Step 6.2.6.2.2

Cancel the common factor.

$f^{- 1} (- \frac{4 x}{7}) = - \frac{4 (- x)}{4 \cdot 1}$

Step 6.2.6.2.3

Rewrite the expression.

$f^{- 1} (- \frac{4 x}{7}) = - \frac{- x}{1}$

Step 6.2.6.2.4

Divide $- x$ by $1$ .

$f^{- 1} (- \frac{4 x}{7}) = x$

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 $f (- \frac{7 x}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{7 x}{4}) = - \frac{4 (- \frac{7 x}{4})}{7}$

Step 6.3.3

Simplify the numerator.

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Step 6.3.3.1

Multiply $- 1$ by $4$ .

$f (- \frac{7 x}{4}) = - \frac{- 4 \frac{7 x}{4}}{7}$

Step 6.3.3.2

Combine $- 4$ and $\frac{7 x}{4}$ .

$f (- \frac{7 x}{4}) = - \frac{\frac{- 4 (7 x)}{4}}{7}$

Step 6.3.4

Multiply $- 4$ by $7$ .

$f (- \frac{7 x}{4}) = - \frac{\frac{- 28 x}{4}}{7}$

Step 6.3.5

Reduce the expression by cancelling the common factors.

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Step 6.3.5.1

Reduce the expression $\frac{- 28 x}{4}$ by cancelling the common factors.

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Step 6.3.5.1.1

Factor $4$ out of $- 28 x$ .

$f (- \frac{7 x}{4}) = - \frac{\frac{4 (- 7 x)}{4}}{7}$

Step 6.3.5.1.2

Factor $4$ out of $4$ .

$f (- \frac{7 x}{4}) = - \frac{\frac{4 (- 7 x)}{4 (1)}}{7}$

Step 6.3.5.1.3

Cancel the common factor.

$f (- \frac{7 x}{4}) = - \frac{\frac{4 (- 7 x)}{4 \cdot 1}}{7}$

Step 6.3.5.1.4

Rewrite the expression.

$f (- \frac{7 x}{4}) = - \frac{\frac{- 7 x}{1}}{7}$

Step 6.3.5.2

Divide $- 7 x$ by $1$ .

$f (- \frac{7 x}{4}) = - \frac{- 7 x}{7}$

Step 6.3.6

Cancel the common factor of $- 7$ and $7$ .

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Step 6.3.6.1

Factor $7$ out of $- 7 x$ .

$f (- \frac{7 x}{4}) = - \frac{7 (- x)}{7}$

Step 6.3.6.2

Cancel the common factors.

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Step 6.3.6.2.1

Factor $7$ out of $7$ .

$f (- \frac{7 x}{4}) = - \frac{7 (- x)}{7 (1)}$

Step 6.3.6.2.2

Cancel the common factor.

$f (- \frac{7 x}{4}) = - \frac{7 (- x)}{7 \cdot 1}$

Step 6.3.6.2.3

Rewrite the expression.

$f (- \frac{7 x}{4}) = - \frac{- x}{1}$

Step 6.3.6.2.4

Divide $- x$ by $1$ .

$f (- \frac{7 x}{4}) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{7 x}{4}$ is the inverse of $f (x) = - \frac{4 x}{7}$ .

$f^{- 1} (x) = - \frac{7 x}{4}$