Finite Math Examples

$25 x + 15 y = 975$

Step 1

Subtract $25 x$ from both sides of the equation.

$15 y = 975 - 25 x$

Step 2

Divide each term in $15 y = 975 - 25 x$ by $15$ and simplify.

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Step 2.1

Divide each term in $15 y = 975 - 25 x$ by $15$ .

$\frac{15 y}{15} = \frac{975}{15} + \frac{- 25 x}{15}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $15$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{15 y}{15} = \frac{975}{15} + \frac{- 25 x}{15}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{975}{15} + \frac{- 25 x}{15}$

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 $975$ by $15$ .

$y = 65 + \frac{- 25 x}{15}$

Step 2.3.1.2

Cancel the common factor of $- 25$ and $15$ .

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Step 2.3.1.2.1

Factor $5$ out of $- 25 x$ .

$y = 65 + \frac{5 (- 5 x)}{15}$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $5$ out of $15$ .

$y = 65 + \frac{5 (- 5 x)}{5 (3)}$

Step 2.3.1.2.2.2

Cancel the common factor.

$y = 65 + \frac{5 (- 5 x)}{5 \cdot 3}$

Step 2.3.1.2.2.3

Rewrite the expression.

$y = 65 + \frac{- 5 x}{3}$

Step 2.3.1.3

Move the negative in front of the fraction.

$y = 65 - \frac{5 x}{3}$

Step 3

Interchange the variables.

$x = 65 - \frac{5 y}{3}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $65 - \frac{5 y}{3} = x$ .

$65 - \frac{5 y}{3} = x$

Step 4.2

Subtract $65$ from both sides of the equation.

$- \frac{5 y}{3} = x - 65$

Step 4.3

Multiply both sides of the equation by $- \frac{3}{5}$ .

$- \frac{3}{5} (- \frac{5 y}{3}) = - \frac{3}{5} (x - 65)$

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 $- \frac{3}{5} (- \frac{5 y}{3})$ .

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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

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$\frac{- 3}{5} (- \frac{5 y}{3}) = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.1.2

Move the leading negative in $- \frac{5 y}{3}$ into the numerator.

$\frac{- 3}{5} \cdot \frac{- 5 y}{3} = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{5} \cdot \frac{- 5 y}{3} = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{5} \cdot \frac{- 5 y}{3} = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{5} (- 5 y) = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.2

Cancel the common factor of $5$ .

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Step 4.4.1.1.2.1

Factor $5$ out of $- 5 y$ .

$\frac{- 1}{5} (5 (- y)) = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{5} (5 (- y)) = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.2.3

Rewrite the expression.

$- - y = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.3

Multiply.

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{5} (x - 65)$

Step 4.4.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{3}{5} (x - 65)$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- \frac{3}{5} (x - 65)$ .

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Step 4.4.2.1.1

Simplify terms.

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Step 4.4.2.1.1.1

Apply the distributive property.

$y = - \frac{3}{5} x - \frac{3}{5} \cdot - 65$

Step 4.4.2.1.1.2

Combine $x$ and $\frac{3}{5}$ .

$y = - \frac{x \cdot 3}{5} - \frac{3}{5} \cdot - 65$

Step 4.4.2.1.1.3

Cancel the common factor of $5$ .

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Step 4.4.2.1.1.3.1

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$y = - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot - 65$

Step 4.4.2.1.1.3.2

Factor $5$ out of $- 65$ .

$y = - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot (5 (- 13))$

Step 4.4.2.1.1.3.3

Cancel the common factor.

$y = - \frac{x \cdot 3}{5} + \frac{- 3}{5} \cdot (5 \cdot - 13)$

Step 4.4.2.1.1.3.4

Rewrite the expression.

$y = - \frac{x \cdot 3}{5} - 3 \cdot - 13$

Step 4.4.2.1.1.4

Multiply $- 3$ by $- 13$ .

$y = - \frac{x \cdot 3}{5} + 39$

Step 4.4.2.1.2

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

$y = - \frac{3 x}{5} + 39$

Step 5

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

$f^{- 1} (x) = - \frac{3 x}{5} + 39$

Step 6

Verify if $f^{- 1} (x) = - \frac{3 x}{5} + 39$ is the inverse of $f (x) = 65 - \frac{5 x}{3}$ .

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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 $f^{- 1} (65 - \frac{5 x}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - \frac{3 (65 - \frac{5 x}{3})}{5} + 39$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Cancel the common factor of $65 - \frac{5 x}{3}$ and $5$ .

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Step 6.2.3.1.1

Factor $5$ out of $3 (65 - \frac{5 x}{3})$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - \frac{5 (3 (13 - \frac{x}{3}))}{5} + 39$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $5$ out of $5$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - \frac{5 (3 (13 - \frac{x}{3}))}{5 (1)} + 39$

Step 6.2.3.1.2.2

Cancel the common factor.

$f^{- 1} (65 - \frac{5 x}{3}) = - \frac{5 (3 (13 - \frac{x}{3}))}{5 \cdot 1} + 39$

Step 6.2.3.1.2.3

Rewrite the expression.

$f^{- 1} (65 - \frac{5 x}{3}) = - \frac{3 (13 - \frac{x}{3})}{1} + 39$

Step 6.2.3.1.2.4

Divide $3 (13 - \frac{x}{3})$ by $1$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - (3 (13 - \frac{x}{3})) + 39$

Step 6.2.3.2

Apply the distributive property.

$f^{- 1} (65 - \frac{5 x}{3}) = - (3 \cdot 13 + 3 (- \frac{x}{3})) + 39$

Step 6.2.3.3

Multiply $3$ by $13$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - (39 + 3 (- \frac{x}{3})) + 39$

Step 6.2.3.4

Cancel the common factor of $3$ .

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Step 6.2.3.4.1

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

$f^{- 1} (65 - \frac{5 x}{3}) = - (39 + 3 (\frac{- x}{3})) + 39$

Step 6.2.3.4.2

Cancel the common factor.

$f^{- 1} (65 - \frac{5 x}{3}) = - (39 + 3 (\frac{- x}{3})) + 39$

Step 6.2.3.4.3

Rewrite the expression.

$f^{- 1} (65 - \frac{5 x}{3}) = - (39 - x) + 39$

Step 6.2.3.5

Apply the distributive property.

$f^{- 1} (65 - \frac{5 x}{3}) = - 1 \cdot 39 + x + 39$

Step 6.2.3.6

Multiply $- 1$ by $39$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - 39 + x + 39$

Step 6.2.3.7

Multiply $- - x$ .

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Step 6.2.3.7.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - 39 + 1 x + 39$

Step 6.2.3.7.2

Multiply $x$ by $1$ .

$f^{- 1} (65 - \frac{5 x}{3}) = - 39 + x + 39$

Step 6.2.4

Combine the opposite terms in $- 39 + x + 39$ .

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Step 6.2.4.1

Add $- 39$ and $39$ .

$f^{- 1} (65 - \frac{5 x}{3}) = x + 0$

Step 6.2.4.2

Add $x$ and $0$ .

$f^{- 1} (65 - \frac{5 x}{3}) = x$

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 $f (- \frac{3 x}{5} + 39)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{3 x}{5} + 39) = 65 - \frac{5 (- \frac{3 x}{5} + 39)}{3}$

Step 6.3.3

Simplify each term.

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Step 6.3.3.1

Cancel the common factor of $- \frac{3 x}{5} + 39$ and $3$ .

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Step 6.3.3.1.1

Factor $3$ out of $5 (- \frac{3 x}{5} + 39)$ .

$f (- \frac{3 x}{5} + 39) = 65 - \frac{3 (5 (- \frac{x}{5} + 13))}{3}$

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 $3$ .

$f (- \frac{3 x}{5} + 39) = 65 - \frac{3 (5 (- \frac{x}{5} + 13))}{3 (1)}$

Step 6.3.3.1.2.2

Cancel the common factor.

$f (- \frac{3 x}{5} + 39) = 65 - \frac{3 (5 (- \frac{x}{5} + 13))}{3 \cdot 1}$

Step 6.3.3.1.2.3

Rewrite the expression.

$f (- \frac{3 x}{5} + 39) = 65 - \frac{5 (- \frac{x}{5} + 13)}{1}$

Step 6.3.3.1.2.4

Divide $5 (- \frac{x}{5} + 13)$ by $1$ .

$f (- \frac{3 x}{5} + 39) = 65 - (5 (- \frac{x}{5} + 13))$

Step 6.3.3.2

Apply the distributive property.

$f (- \frac{3 x}{5} + 39) = 65 - (5 (- \frac{x}{5}) + 5 \cdot 13)$

Step 6.3.3.3

Cancel the common factor of $5$ .

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Step 6.3.3.3.1

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

$f (- \frac{3 x}{5} + 39) = 65 - (5 (\frac{- x}{5}) + 5 \cdot 13)$

Step 6.3.3.3.2

Cancel the common factor.

$f (- \frac{3 x}{5} + 39) = 65 - (5 (\frac{- x}{5}) + 5 \cdot 13)$

Step 6.3.3.3.3

Rewrite the expression.

$f (- \frac{3 x}{5} + 39) = 65 - (- x + 5 \cdot 13)$

Step 6.3.3.4

Multiply $5$ by $13$ .

$f (- \frac{3 x}{5} + 39) = 65 - (- x + 65)$

Step 6.3.3.5

Apply the distributive property.

$f (- \frac{3 x}{5} + 39) = 65 + x - 1 \cdot 65$

Step 6.3.3.6

Multiply $- - x$ .

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Step 6.3.3.6.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 x}{5} + 39) = 65 + 1 x - 1 \cdot 65$

Step 6.3.3.6.2

Multiply $x$ by $1$ .

$f (- \frac{3 x}{5} + 39) = 65 + x - 1 \cdot 65$

Step 6.3.3.7

Multiply $- 1$ by $65$ .

$f (- \frac{3 x}{5} + 39) = 65 + x - 65$

Step 6.3.4

Combine the opposite terms in $65 + x - 65$ .

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Step 6.3.4.1

Subtract $65$ from $65$ .

$f (- \frac{3 x}{5} + 39) = x + 0$

Step 6.3.4.2

Add $x$ and $0$ .

$f (- \frac{3 x}{5} + 39) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{3 x}{5} + 39$ is the inverse of $f (x) = 65 - \frac{5 x}{3}$ .

$f^{- 1} (x) = - \frac{3 x}{5} + 39$