Finite Math Examples

$6 x - 7 y - 3 = 0$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Subtract $6 x$ from both sides of the equation.

$- 7 y - 3 = - 6 x$

Step 1.2

Add $3$ to both sides of the equation.

$- 7 y = - 6 x + 3$

Step 2

Divide each term in $- 7 y = - 6 x + 3$ by $- 7$ and simplify.

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Step 2.1

Divide each term in $- 7 y = - 6 x + 3$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{- 6 x}{- 7} + \frac{3}{- 7}$

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.

$\frac{- 7 y}{- 7} = \frac{- 6 x}{- 7} + \frac{3}{- 7}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 6 x}{- 7} + \frac{3}{- 7}$

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

Dividing two negative values results in a positive value.

$y = \frac{6 x}{7} + \frac{3}{- 7}$

Step 2.3.1.2

Move the negative in front of the fraction.

$y = \frac{6 x}{7} - \frac{3}{7}$

Step 3

Interchange the variables.

$x = \frac{6 y}{7} - \frac{3}{7}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $\frac{6 y}{7} - \frac{3}{7} = x$ .

$\frac{6 y}{7} - \frac{3}{7} = x$

Step 4.2

Add $\frac{3}{7}$ to both sides of the equation.

$\frac{6 y}{7} = x + \frac{3}{7}$

Step 4.3

Multiply both sides of the equation by $\frac{7}{6}$ .

$\frac{7}{6} \cdot \frac{6 y}{7} = \frac{7}{6} (x + \frac{3}{7})$

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 $\frac{7}{6} \cdot \frac{6 y}{7}$ .

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Step 4.4.1.1.1

Cancel the common factor of $7$ .

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Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{7}{6} \cdot \frac{6 y}{7} = \frac{7}{6} (x + \frac{3}{7})$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{6} (6 y) = \frac{7}{6} (x + \frac{3}{7})$

Step 4.4.1.1.2

Cancel the common factor of $6$ .

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Step 4.4.1.1.2.1

Factor $6$ out of $6 y$ .

$\frac{1}{6} (6 (y)) = \frac{7}{6} (x + \frac{3}{7})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{6} (6 y) = \frac{7}{6} (x + \frac{3}{7})$

Step 4.4.1.1.2.3

Rewrite the expression.

$y = \frac{7}{6} (x + \frac{3}{7})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $\frac{7}{6} (x + \frac{3}{7})$ .

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Step 4.4.2.1.1

Apply the distributive property.

$y = \frac{7}{6} x + \frac{7}{6} \cdot \frac{3}{7}$

Step 4.4.2.1.2

Combine $\frac{7}{6}$ and $x$ .

$y = \frac{7 x}{6} + \frac{7}{6} \cdot \frac{3}{7}$

Step 4.4.2.1.3

Cancel the common factor of $7$ .

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Step 4.4.2.1.3.1

Cancel the common factor.

$y = \frac{7 x}{6} + \frac{7}{6} \cdot \frac{3}{7}$

Step 4.4.2.1.3.2

Rewrite the expression.

$y = \frac{7 x}{6} + \frac{1}{6} \cdot 3$

Step 4.4.2.1.4

Cancel the common factor of $3$ .

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Step 4.4.2.1.4.1

Factor $3$ out of $6$ .

$y = \frac{7 x}{6} + \frac{1}{3 (2)} \cdot 3$

Step 4.4.2.1.4.2

Cancel the common factor.

$y = \frac{7 x}{6} + \frac{1}{3 \cdot 2} \cdot 3$

Step 4.4.2.1.4.3

Rewrite the expression.

$y = \frac{7 x}{6} + \frac{1}{2}$

Step 5

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

$f^{- 1} (x) = \frac{7 x}{6} + \frac{1}{2}$

Step 6

Verify if $f^{- 1} (x) = \frac{7 x}{6} + \frac{1}{2}$ is the inverse of $f (x) = \frac{6 x}{7} - \frac{3}{7}$ .

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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 $f^{- 1} (\frac{6 x}{7} - \frac{3}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{7 (\frac{6 x}{7} - \frac{3}{7})}{6} + \frac{1}{2}$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{7 (\frac{6 x - 3}{7})}{6} + \frac{1}{2}$

Step 6.2.3.1.2

Factor $3$ out of $6 x - 3$ .

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Step 6.2.3.1.2.1

Factor $3$ out of $6 x$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{7 (\frac{3 (2 x) - 3}{7})}{6} + \frac{1}{2}$

Step 6.2.3.1.2.2

Factor $3$ out of $- 3$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{7 (\frac{3 (2 x) + 3 (- 1)}{7})}{6} + \frac{1}{2}$

Step 6.2.3.1.2.3

Factor $3$ out of $3 (2 x) + 3 (- 1)$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{7 (\frac{3 (2 x - 1)}{7})}{6} + \frac{1}{2}$

Step 6.2.3.2

Combine $7$ and $\frac{3 (2 x - 1)}{7}$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{\frac{7 (3 (2 x - 1))}{7}}{6} + \frac{1}{2}$

Step 6.2.3.3

Multiply $7$ by $3$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{\frac{21 (2 x - 1)}{7}}{6} + \frac{1}{2}$

Step 6.2.3.4

Reduce the expression by cancelling the common factors.

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Step 6.2.3.4.1

Reduce the expression $\frac{21 (2 x - 1)}{7}$ by cancelling the common factors.

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Step 6.2.3.4.1.1

Factor $7$ out of $21 (2 x - 1)$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{\frac{7 (3 (2 x - 1))}{7}}{6} + \frac{1}{2}$

Step 6.2.3.4.1.2

Factor $7$ out of $7$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{\frac{7 (3 (2 x - 1))}{7 (1)}}{6} + \frac{1}{2}$

Step 6.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{\frac{7 (3 (2 x - 1))}{7 \cdot 1}}{6} + \frac{1}{2}$

Step 6.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{\frac{3 (2 x - 1)}{1}}{6} + \frac{1}{2}$

Step 6.2.3.4.2

Divide $3 (2 x - 1)$ by $1$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{3 (2 x - 1)}{6} + \frac{1}{2}$

Step 6.2.3.5

Cancel the common factors.

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Step 6.2.3.5.1

Factor $3$ out of $6$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{3 (2 x - 1)}{3 \cdot 2} + \frac{1}{2}$

Step 6.2.3.5.2

Cancel the common factor.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{3 (2 x - 1)}{3 \cdot 2} + \frac{1}{2}$

Step 6.2.3.5.3

Rewrite the expression.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{2 x - 1}{2} + \frac{1}{2}$

Step 6.2.4

Simplify terms.

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Step 6.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{2 x - 1 + 1}{2}$

Step 6.2.4.2

Combine the opposite terms in $2 x - 1 + 1$ .

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Step 6.2.4.2.1

Add $- 1$ and $1$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{2 x + 0}{2}$

Step 6.2.4.2.2

Add $2 x$ and $0$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{2 x}{2}$

Step 6.2.4.3

Cancel the common factor of $2$ .

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Step 6.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = \frac{2 x}{2}$

Step 6.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{6 x}{7} - \frac{3}{7}) = x$

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 $f (\frac{7 x}{6} + \frac{1}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{6 (\frac{7 x}{6} + \frac{1}{2})}{7} - \frac{3}{7}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{6 (\frac{7 x}{6} + \frac{1}{2}) - 3}{7}$

Step 6.3.4

Simplify each term.

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Step 6.3.4.1

Apply the distributive property.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{6 (\frac{7 x}{6}) + 6 (\frac{1}{2}) - 3}{7}$

Step 6.3.4.2

Cancel the common factor of $6$ .

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Step 6.3.4.2.1

Cancel the common factor.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{6 (\frac{7 x}{6}) + 6 (\frac{1}{2}) - 3}{7}$

Step 6.3.4.2.2

Rewrite the expression.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x + 6 (\frac{1}{2}) - 3}{7}$

Step 6.3.4.3

Cancel the common factor of $2$ .

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Step 6.3.4.3.1

Factor $2$ out of $6$ .

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x + 2 (3) (\frac{1}{2}) - 3}{7}$

Step 6.3.4.3.2

Cancel the common factor.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x + 2 \cdot (3 (\frac{1}{2})) - 3}{7}$

Step 6.3.4.3.3

Rewrite the expression.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x + 3 - 3}{7}$

Step 6.3.5

Simplify terms.

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Step 6.3.5.1

Combine the opposite terms in $7 x + 3 - 3$ .

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Step 6.3.5.1.1

Subtract $3$ from $3$ .

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x + 0}{7}$

Step 6.3.5.1.2

Add $7 x$ and $0$ .

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x}{7}$

Step 6.3.5.2

Cancel the common factor of $7$ .

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Step 6.3.5.2.1

Cancel the common factor.

$f (\frac{7 x}{6} + \frac{1}{2}) = \frac{7 x}{7}$

Step 6.3.5.2.2

Divide $x$ by $1$ .

$f (\frac{7 x}{6} + \frac{1}{2}) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{6} + \frac{1}{2}$ is the inverse of $f (x) = \frac{6 x}{7} - \frac{3}{7}$ .

$f^{- 1} (x) = \frac{7 x}{6} + \frac{1}{2}$