Find the Inverse y=-5/8x+10

$y = - \frac{5}{8} x + 10$

Step 1

Interchange the variables.

$x = - \frac{5}{8} y + 10$

Step 2

Solve for $y$ .

Rewrite the equation as $- \frac{5}{8} y + 10 = x$ .

$- \frac{5}{8} y + 10 = x$

Simplify each term.

Combine $y$ and $\frac{5}{8}$ .

$- \frac{y \cdot 5}{8} + 10 = x$

Move $5$ to the left of $y$ .

$- \frac{5 y}{8} + 10 = x$

Subtract $10$ from both sides of the equation.

$- \frac{5 y}{8} = x - 10$

Multiply both sides of the equation by $- \frac{8}{5}$ .

$- \frac{8}{5} (- \frac{5 y}{8}) = - \frac{8}{5} (x - 10)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{8}{5} (- \frac{5 y}{8})$ .

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{8}{5}$ into the numerator.

$\frac{- 8}{5} (- \frac{5 y}{8}) = - \frac{8}{5} (x - 10)$

Move the leading negative in $- \frac{5 y}{8}$ into the numerator.

$\frac{- 8}{5} \cdot \frac{- 5 y}{8} = - \frac{8}{5} (x - 10)$

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{5} \cdot \frac{- 5 y}{8} = - \frac{8}{5} (x - 10)$

Cancel the common factor.

$\frac{8 \cdot - 1}{5} \cdot \frac{- 5 y}{8} = - \frac{8}{5} (x - 10)$

Rewrite the expression.

$\frac{- 1}{5} (- 5 y) = - \frac{8}{5} (x - 10)$

Cancel the common factor of $5$ .

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

$\frac{- 1}{5} (5 (- y)) = - \frac{8}{5} (x - 10)$

Cancel the common factor.

$\frac{- 1}{5} (5 (- y)) = - \frac{8}{5} (x - 10)$

Rewrite the expression.

$- - y = - \frac{8}{5} (x - 10)$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{8}{5} (x - 10)$

Multiply $y$ by $1$ .

$y = - \frac{8}{5} (x - 10)$

Simplify the right side.

Simplify $- \frac{8}{5} (x - 10)$ .

Simplify terms.

Apply the distributive property.

$y = - \frac{8}{5} x - \frac{8}{5} \cdot - 10$

Combine $x$ and $\frac{8}{5}$ .

$y = - \frac{x \cdot 8}{5} - \frac{8}{5} \cdot - 10$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{8}{5}$ into the numerator.

$y = - \frac{x \cdot 8}{5} + \frac{- 8}{5} \cdot - 10$

Factor $5$ out of $- 10$ .

$y = - \frac{x \cdot 8}{5} + \frac{- 8}{5} \cdot (5 (- 2))$

Cancel the common factor.

$y = - \frac{x \cdot 8}{5} + \frac{- 8}{5} \cdot (5 \cdot - 2)$

Rewrite the expression.

$y = - \frac{x \cdot 8}{5} - 8 \cdot - 2$

Multiply $- 8$ by $- 2$ .

$y = - \frac{x \cdot 8}{5} + 16$

Move $8$ to the left of $x$ .

$y = - \frac{8 x}{5} + 16$

Step 3

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

$f^{- 1} (x) = - \frac{8 x}{5} + 16$

Step 4

Verify if $f^{- 1} (x) = - \frac{8 x}{5} + 16$ is the inverse of $f (x) = - \frac{5}{8} x + 10$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (- \frac{5}{8} x + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 (- \frac{5}{8} x + 10)}{5} + 16$

Simplify each term.

Simplify the numerator.

Combine $x$ and $\frac{5}{8}$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 (- \frac{x \cdot 5}{8} + 10)}{5} + 16$

Move $5$ to the left of $x$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 (- \frac{5 \cdot x}{8} + 10)}{5} + 16$

Factor $5$ out of $- \frac{5 x}{8} + 10$ .

Factor $5$ out of $- \frac{5 x}{8}$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 (5 (- \frac{x}{8}) + 10)}{5} + 16$

Factor $5$ out of $10$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 (5 (- \frac{x}{8}) + 5 (2))}{5} + 16$

Factor $5$ out of $5 (- \frac{x}{8}) + 5 (2)$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 (5 (- \frac{x}{8} + 2))}{5} + 16$

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{8 \cdot (5 (- \frac{x}{8} + 2))}{5} + 16$

Multiply $8$ by $5$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{40 (- \frac{x}{8} + 2)}{5} + 16$

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{40 (- \frac{x}{8} + 2 \cdot \frac{8}{8})}{5} + 16$

Combine $2$ and $\frac{8}{8}$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{40 (- \frac{x}{8} + \frac{2 \cdot 8}{8})}{5} + 16$

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{40 (\frac{- x + 2 \cdot 8}{8})}{5} + 16$

Multiply $2$ by $8$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{40 (\frac{- x + 16}{8})}{5} + 16$

Combine $40$ and $\frac{- x + 16}{8}$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{\frac{40 (- x + 16)}{8}}{5} + 16$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{40 (- x + 16)}{8}$ by cancelling the common factors.

Factor $8$ out of $40 (- x + 16)$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{\frac{8 (5 (- x + 16))}{8}}{5} + 16$

Factor $8$ out of $8$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{\frac{8 (5 (- x + 16))}{8 (1)}}{5} + 16$

Cancel the common factor.

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{\frac{8 (5 (- x + 16))}{8 \cdot 1}}{5} + 16$

Rewrite the expression.

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{\frac{5 (- x + 16)}{1}}{5} + 16$

Divide $5 (- x + 16)$ by $1$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{5 (- x + 16)}{5} + 16$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (- \frac{5}{8} x + 10) = - \frac{5 (- x + 16)}{5} + 16$

Divide $- x + 16$ by $1$ .

$f^{- 1} (- \frac{5}{8} x + 10) = - (- x + 16) + 16$

Apply the distributive property.

$f^{- 1} (- \frac{5}{8} x + 10) = x - 1 \cdot 16 + 16$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{5}{8} x + 10) = 1 x - 1 \cdot 16 + 16$

Multiply $x$ by $1$ .

$f^{- 1} (- \frac{5}{8} x + 10) = x - 1 \cdot 16 + 16$

Multiply $- 1$ by $16$ .

$f^{- 1} (- \frac{5}{8} x + 10) = x - 16 + 16$

Combine the opposite terms in $x - 16 + 16$ .

Add $- 16$ and $16$ .

$f^{- 1} (- \frac{5}{8} x + 10) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (- \frac{5}{8} x + 10) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{8 x}{5} + 16)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{8 x}{5} + 16) = - \frac{5}{8} \cdot (- \frac{8 x}{5} + 16) + 10$

Simplify each term.

Apply the distributive property.

$f (- \frac{8 x}{5} + 16) = - \frac{5}{8} \cdot (- \frac{8 x}{5}) - \frac{5}{8} \cdot 16 + 10$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$f (- \frac{8 x}{5} + 16) = \frac{- 5}{8} \cdot (- \frac{8 x}{5}) - \frac{5}{8} \cdot 16 + 10$

Move the leading negative in $- \frac{8 x}{5}$ into the numerator.

$f (- \frac{8 x}{5} + 16) = \frac{- 5}{8} \cdot \frac{- 8 x}{5} - \frac{5}{8} \cdot 16 + 10$

Factor $5$ out of $- 5$ .

$f (- \frac{8 x}{5} + 16) = \frac{5 (- 1)}{8} \cdot \frac{- 8 x}{5} - \frac{5}{8} \cdot 16 + 10$

Cancel the common factor.

$f (- \frac{8 x}{5} + 16) = \frac{5 \cdot - 1}{8} \cdot \frac{- 8 x}{5} - \frac{5}{8} \cdot 16 + 10$

Rewrite the expression.

$f (- \frac{8 x}{5} + 16) = \frac{- 1}{8} \cdot (- 8 x) - \frac{5}{8} \cdot 16 + 10$

Cancel the common factor of $8$ .

Factor $8$ out of $- 8 x$ .

$f (- \frac{8 x}{5} + 16) = \frac{- 1}{8} \cdot (8 (- x)) - \frac{5}{8} \cdot 16 + 10$

Cancel the common factor.

$f (- \frac{8 x}{5} + 16) = \frac{- 1}{8} \cdot (8 (- x)) - \frac{5}{8} \cdot 16 + 10$

Rewrite the expression.

$f (- \frac{8 x}{5} + 16) = - 1 (- x) - \frac{5}{8} \cdot 16 + 10$

Multiply $- 1$ by $- 1$ .

$f (- \frac{8 x}{5} + 16) = 1 x - \frac{5}{8} \cdot 16 + 10$

Multiply $x$ by $1$ .

$f (- \frac{8 x}{5} + 16) = x - \frac{5}{8} \cdot 16 + 10$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$f (- \frac{8 x}{5} + 16) = x + \frac{- 5}{8} \cdot 16 + 10$

Factor $8$ out of $16$ .

$f (- \frac{8 x}{5} + 16) = x + \frac{- 5}{8} \cdot (8 (2)) + 10$

Cancel the common factor.

$f (- \frac{8 x}{5} + 16) = x + \frac{- 5}{8} \cdot (8 \cdot 2) + 10$

Rewrite the expression.

$f (- \frac{8 x}{5} + 16) = x - 5 \cdot 2 + 10$

Multiply $- 5$ by $2$ .

$f (- \frac{8 x}{5} + 16) = x - 10 + 10$

Combine the opposite terms in $x - 10 + 10$ .

Add $- 10$ and $10$ .

$f (- \frac{8 x}{5} + 16) = x + 0$

Add $x$ and $0$ .

$f (- \frac{8 x}{5} + 16) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{8 x}{5} + 16$ is the inverse of $f (x) = - \frac{5}{8} x + 10$ .

$f^{- 1} (x) = - \frac{8 x}{5} + 16$

Find the Inverse y=-5/8x+10

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