Finite Math Examples

4 x + 7 y

$4 x + 7 y$

Step 1

Subtract

4 x

$4 x$ from both sides of the equation.

7 y = - 4 x

$7 y = - 4 x$

Step 2

Divide each term in

7 y = - 4 x

$7 y = - 4 x$ by

7

$7$ and simplify.

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Step 2.1

Divide each term in

7 y = - 4 x

$7 y = - 4 x$ by

7

$7$ .

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

$7$ .

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Step 2.2.1.1

Cancel the common factor.

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

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

Step 2.2.1.2

Divide

y

$y$ by

1

$1$ .

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

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

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

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

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

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

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}

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

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

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

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

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

Step 3

Interchange the variables.

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

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

Step 4

Solve for

y

$y$ .

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Step 4.1

Rewrite the equation as

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

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

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

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

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})

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

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Step 4.3.1.1.1

Cancel the common factor of

7

$7$ .

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Step 4.3.1.1.1.1

Move the leading negative in

- \frac{7}{4}

$- \frac{7}{4}$ into the numerator.

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

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

Step 4.3.1.1.1.2

Move the leading negative in

- \frac{4 y}{7}

$- \frac{4 y}{7}$ into the numerator.

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

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

Step 4.3.1.1.1.3

Factor

7

$7$ out of

- 7

$- 7$ .

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

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

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

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

Step 4.3.1.1.1.5

Rewrite the expression.

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

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

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

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

Step 4.3.1.1.2

Cancel the common factor of

4

$4$ .

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Step 4.3.1.1.2.1

Factor

4

$4$ out of

- 4 y

$- 4 y$ .

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

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

Step 4.3.1.1.2.2

Cancel the common factor.

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

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

Step 4.3.1.1.2.3

Rewrite the expression.

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

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

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

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

Step 4.3.1.1.3

Multiply.

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Step 4.3.1.1.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 4.3.1.1.3.2

Multiply

y

$y$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify

- \frac{7}{4} x

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

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Step 4.3.2.1.1

Combine

x

$x$ and

\frac{7}{4}

$\frac{7}{4}$ .

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

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

Step 4.3.2.1.2

Move

7

$7$ to the left of

x

$x$ .

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

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

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

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

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

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

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

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

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

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

Step 5

Replace

y

$y$ with

f^{- 1} (x)

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

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

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

Step 6

Verify if

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

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

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

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

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Step 6.1

To verify the inverse, check if

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

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

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

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

Step 6.2

Evaluate

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

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

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Step 6.2.1

Set up the composite result function.

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

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

Step 6.2.2

Evaluate

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

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

f

$f$ into

f^{- 1}

$f^{- 1}$ .

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

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

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

Multiply

- 1

$- 1$ by

7

$7$ .

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

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

Step 6.2.3.2

Combine

- 7

$- 7$ and

\frac{4 x}{7}

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

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

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

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

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

Step 6.2.4

Multiply

- 7

$- 7$ by

4

$4$ .

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

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

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Step 6.2.5.1.1

Factor

7

$7$ out of

- 28 x

$- 28 x$ .

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

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

Step 6.2.5.1.2

Factor

7

$7$ out of

7

$7$ .

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

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

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}

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

Step 6.2.5.1.4

Rewrite the expression.

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

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

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

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

Step 6.2.5.2

Divide

- 4 x

$- 4 x$ by

1

$1$ .

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

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

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

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

Step 6.2.6

Cancel the common factor of

- 4

$- 4$ and

4

$4$ .

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Step 6.2.6.1

Factor

4

$4$ out of

- 4 x

$- 4 x$ .

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

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

Step 6.2.6.2

Cancel the common factors.

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Step 6.2.6.2.1

Factor

4

$4$ out of

4

$4$ .

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

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

Step 6.2.6.2.2

Cancel the common factor.

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

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

Step 6.2.6.2.3

Rewrite the expression.

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

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

Step 6.2.6.2.4

Divide

- x

$- x$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 6.3

Evaluate

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

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

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Step 6.3.1

Set up the composite result function.

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

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

Step 6.3.2

Evaluate

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

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

f^{- 1}

$f^{- 1}$ into

f

$f$ .

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

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

Step 6.3.3

Simplify the numerator.

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Step 6.3.3.1

Multiply

- 1

$- 1$ by

4

$4$ .

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

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

Step 6.3.3.2

Combine

- 4

$- 4$ and

\frac{7 x}{4}

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

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

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

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

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

Step 6.3.4

Multiply

- 4

$- 4$ by

7

$7$ .

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

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

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Step 6.3.5.1.1

Factor

4

$4$ out of

- 28 x

$- 28 x$ .

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

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

Step 6.3.5.1.2

Factor

4

$4$ out of

4

$4$ .

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

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

Step 6.3.5.1.3

Cancel the common factor.

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

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

Step 6.3.5.1.4

Rewrite the expression.

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

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

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

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

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