Finite Math Examples

$2 x - 7 y = 24$

Step 1

Subtract $2 x$ from both sides of the equation.

$- 7 y = 24 - 2 x$

Step 2

Divide each term in $- 7 y = 24 - 2 x$ by $- 7$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- 7 y = 24 - 2 x$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{24}{- 7} + \frac{- 2 x}{- 7}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $- 7$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{24}{- 7} + \frac{- 2 x}{- 7}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{24}{- 7} + \frac{- 2 x}{- 7}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{24}{7} + \frac{- 2 x}{- 7}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{24}{7} + \frac{2 x}{7}$

Step 3

Interchange the variables.

$x = - \frac{24}{7} + \frac{2 y}{7}$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $- \frac{24}{7} + \frac{2 y}{7} = x$ .

$- \frac{24}{7} + \frac{2 y}{7} = x$

Step 4.2

Add $\frac{24}{7}$ to both sides of the equation.

$\frac{2 y}{7} = x + \frac{24}{7}$

Step 4.3

Multiply both sides of the equation by $\frac{7}{2}$ .

$\frac{7}{2} \cdot \frac{2 y}{7} = \frac{7}{2} (x + \frac{24}{7})$

Step 4.4

Simplify both sides of the equation.

Tap for more steps...

Step 4.4.1

Simplify the left side.

Tap for more steps...

Step 4.4.1.1

Simplify $\frac{7}{2} \cdot \frac{2 y}{7}$ .

Tap for more steps...

Step 4.4.1.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{7}{2} \cdot \frac{2 y}{7} = \frac{7}{2} (x + \frac{24}{7})$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{7}{2} (x + \frac{24}{7})$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{7}{2} (x + \frac{24}{7})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{7}{2} (x + \frac{24}{7})$

Step 4.4.1.1.2.3

Rewrite the expression.

$y = \frac{7}{2} (x + \frac{24}{7})$

Step 4.4.2

Simplify the right side.

Tap for more steps...

Step 4.4.2.1

Simplify $\frac{7}{2} (x + \frac{24}{7})$ .

Tap for more steps...

Step 4.4.2.1.1

Apply the distributive property.

$y = \frac{7}{2} x + \frac{7}{2} \cdot \frac{24}{7}$

Step 4.4.2.1.2

Combine $\frac{7}{2}$ and $x$ .

$y = \frac{7 x}{2} + \frac{7}{2} \cdot \frac{24}{7}$

Step 4.4.2.1.3

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.4.2.1.3.1

Cancel the common factor.

$y = \frac{7 x}{2} + \frac{7}{2} \cdot \frac{24}{7}$

Step 4.4.2.1.3.2

Rewrite the expression.

$y = \frac{7 x}{2} + \frac{1}{2} \cdot 24$

Step 4.4.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.2.1.4.1

Factor $2$ out of $24$ .

$y = \frac{7 x}{2} + \frac{1}{2} \cdot (2 (12))$

Step 4.4.2.1.4.2

Cancel the common factor.

$y = \frac{7 x}{2} + \frac{1}{2} \cdot (2 \cdot 12)$

Step 4.4.2.1.4.3

Rewrite the expression.

$y = \frac{7 x}{2} + 12$

Step 5

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

$f^{- 1} (x) = \frac{7 x}{2} + 12$

Step 6

Verify if $f^{- 1} (x) = \frac{7 x}{2} + 12$ is the inverse of $f (x) = - \frac{24}{7} + \frac{2 x}{7}$ .

Tap for more steps...

Step 6.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 6.2

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

Tap for more steps...

Step 6.2.1

Set up the composite result function.

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

Step 6.2.2

Evaluate $f^{- 1} (- \frac{24}{7} + \frac{2 x}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{7 (- \frac{24}{7} + \frac{2 x}{7})}{2} + 12$

Step 6.2.3

Simplify each term.

Tap for more steps...

Step 6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 6.2.3.1.1

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{7 (\frac{- 24 + 2 x}{7})}{2} + 12$

Step 6.2.3.1.2

Reorder $- 24$ and $2 x$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{7 (\frac{2 x - 24}{7})}{2} + 12$

Step 6.2.3.1.3

Factor $2$ out of $2 x - 24$ .

Tap for more steps...

Step 6.2.3.1.3.1

Factor $2$ out of $2 x$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{7 (\frac{2 (x) - 24}{7})}{2} + 12$

Step 6.2.3.1.3.2

Factor $2$ out of $- 24$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{7 (\frac{2 x + 2 \cdot - 12}{7})}{2} + 12$

Step 6.2.3.1.3.3

Factor $2$ out of $2 x + 2 \cdot - 12$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{7 (\frac{2 (x - 12)}{7})}{2} + 12$

Step 6.2.3.2

Combine $7$ and $\frac{2 (x - 12)}{7}$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{\frac{7 (2 (x - 12))}{7}}{2} + 12$

Step 6.2.3.3

Multiply $7$ by $2$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{\frac{14 (x - 12)}{7}}{2} + 12$

Step 6.2.3.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.2.3.4.1

Reduce the expression $\frac{14 (x - 12)}{7}$ by cancelling the common factors.

Tap for more steps...

Step 6.2.3.4.1.1

Factor $7$ out of $14 (x - 12)$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{\frac{7 (2 (x - 12))}{7}}{2} + 12$

Step 6.2.3.4.1.2

Factor $7$ out of $7$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{\frac{7 (2 (x - 12))}{7 (1)}}{2} + 12$

Step 6.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{\frac{7 (2 (x - 12))}{7 \cdot 1}}{2} + 12$

Step 6.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{\frac{2 (x - 12)}{1}}{2} + 12$

Step 6.2.3.4.2

Divide $2 (x - 12)$ by $1$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{2 (x - 12)}{2} + 12$

Step 6.2.3.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.3.5.1

Cancel the common factor.

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = \frac{2 (x - 12)}{2} + 12$

Step 6.2.3.5.2

Divide $x - 12$ by $1$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = x - 12 + 12$

Step 6.2.4

Combine the opposite terms in $x - 12 + 12$ .

Tap for more steps...

Step 6.2.4.1

Add $- 12$ and $12$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = x + 0$

Step 6.2.4.2

Add $x$ and $0$ .

$f^{- 1} (- \frac{24}{7} + \frac{2 x}{7}) = x$

Step 6.3

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

Tap for more steps...

Step 6.3.1

Set up the composite result function.

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

Step 6.3.2

Evaluate $f (\frac{7 x}{2} + 12)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{2} + 12) = - \frac{24}{7} + \frac{2 (\frac{7 x}{2} + 12)}{7}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (\frac{7 x}{2} + 12) = \frac{- 24 + 2 (\frac{7 x}{2} + 12)}{7}$

Step 6.3.4

Simplify each term.

Tap for more steps...

Step 6.3.4.1

Apply the distributive property.

$f (\frac{7 x}{2} + 12) = \frac{- 24 + 2 (\frac{7 x}{2}) + 2 \cdot 12}{7}$

Step 6.3.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.4.2.1

Cancel the common factor.

$f (\frac{7 x}{2} + 12) = \frac{- 24 + 2 (\frac{7 x}{2}) + 2 \cdot 12}{7}$

Step 6.3.4.2.2

Rewrite the expression.

$f (\frac{7 x}{2} + 12) = \frac{- 24 + 7 x + 2 \cdot 12}{7}$

Step 6.3.4.3

Multiply $2$ by $12$ .

$f (\frac{7 x}{2} + 12) = \frac{- 24 + 7 x + 24}{7}$

Step 6.3.5

Simplify terms.

Tap for more steps...

Step 6.3.5.1

Combine the opposite terms in $- 24 + 7 x + 24$ .

Tap for more steps...

Step 6.3.5.1.1

Add $- 24$ and $24$ .

$f (\frac{7 x}{2} + 12) = \frac{7 x + 0}{7}$

Step 6.3.5.1.2

Add $7 x$ and $0$ .

$f (\frac{7 x}{2} + 12) = \frac{7 x}{7}$

Step 6.3.5.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 6.3.5.2.1

Cancel the common factor.

$f (\frac{7 x}{2} + 12) = \frac{7 x}{7}$

Step 6.3.5.2.2

Divide $x$ by $1$ .

$f (\frac{7 x}{2} + 12) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{2} + 12$ is the inverse of $f (x) = - \frac{24}{7} + \frac{2 x}{7}$ .

$f^{- 1} (x) = \frac{7 x}{2} + 12$