Find the Inverse y=-3/7x+5/7

$y = - \frac{3}{7} x + \frac{5}{7}$

Interchange the variables.

$x = - \frac{3}{7} y + \frac{5}{7}$

Solve for $y$ .

Rewrite the equation as $- \frac{3}{7} y + \frac{5}{7} = x$ .

$- \frac{3}{7} y + \frac{5}{7} = x$

Simplify each term.

Combine $y$ and $\frac{3}{7}$ .

$- \frac{y \cdot 3}{7} + \frac{5}{7} = x$

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

$- \frac{3 y}{7} + \frac{5}{7} = x$

Subtract $\frac{5}{7}$ from both sides of the equation.

$- \frac{3 y}{7} = x - \frac{5}{7}$

Multiply both sides of the equation by $- \frac{7}{3}$ .

$- \frac{7}{3} (- \frac{3 y}{7}) = - \frac{7}{3} (x - \frac{5}{7})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{7}{3} (- \frac{3 y}{7})$ .

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$\frac{- 7}{3} (- \frac{3 y}{7}) = - \frac{7}{3} (x - \frac{5}{7})$

Move the leading negative in $- \frac{3 y}{7}$ into the numerator.

$\frac{- 7}{3} \cdot \frac{- 3 y}{7} = - \frac{7}{3} (x - \frac{5}{7})$

Factor $7$ out of $- 7$ .

$\frac{7 (- 1)}{3} \cdot \frac{- 3 y}{7} = - \frac{7}{3} (x - \frac{5}{7})$

Cancel the common factor.

$\frac{7 \cdot - 1}{3} \cdot \frac{- 3 y}{7} = - \frac{7}{3} (x - \frac{5}{7})$

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{7}{3} (x - \frac{5}{7})$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{7}{3} (x - \frac{5}{7})$

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{7}{3} (x - \frac{5}{7})$

Rewrite the expression.

$- - y = - \frac{7}{3} (x - \frac{5}{7})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{7}{3} (x - \frac{5}{7})$

Multiply $y$ by $1$ .

$y = - \frac{7}{3} (x - \frac{5}{7})$

Simplify the right side.

Simplify $- \frac{7}{3} (x - \frac{5}{7})$ .

Simplify terms.

Apply the distributive property.

$y = - \frac{7}{3} x - \frac{7}{3} (- \frac{5}{7})$

Combine $x$ and $\frac{7}{3}$ .

$y = - \frac{x \cdot 7}{3} - \frac{7}{3} (- \frac{5}{7})$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$y = - \frac{x \cdot 7}{3} + \frac{- 7}{3} (- \frac{5}{7})$

Move the leading negative in $- \frac{5}{7}$ into the numerator.

$y = - \frac{x \cdot 7}{3} + \frac{- 7}{3} \cdot \frac{- 5}{7}$

Factor $7$ out of $- 7$ .

$y = - \frac{x \cdot 7}{3} + \frac{7 (- 1)}{3} \cdot \frac{- 5}{7}$

Cancel the common factor.

$y = - \frac{x \cdot 7}{3} + \frac{7 \cdot - 1}{3} \cdot \frac{- 5}{7}$

Rewrite the expression.

$y = - \frac{x \cdot 7}{3} + \frac{- 1}{3} \cdot - 5$

Combine $\frac{- 1}{3}$ and $- 5$ .

$y = - \frac{x \cdot 7}{3} + \frac{- - 5}{3}$

Multiply $- 1$ by $- 5$ .

$y = - \frac{x \cdot 7}{3} + \frac{5}{3}$

Move $7$ to the left of $x$ .

$y = - \frac{7 x}{3} + \frac{5}{3}$

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

$f^{- 1} (x) = - \frac{7 x}{3} + \frac{5}{3}$

Verify if $f^{- 1} (x) = - \frac{7 x}{3} + \frac{5}{3}$ is the inverse of $f (x) = - \frac{3}{7} x + \frac{5}{7}$ .

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 $f^{- 1} (- \frac{3}{7} x + \frac{5}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = - \frac{7 (- \frac{3}{7} x + \frac{5}{7})}{3} + \frac{5}{3}$

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{- 7 (- \frac{3}{7} x + \frac{5}{7}) + 5}{3}$

Simplify each term.

Combine $x$ and $\frac{3}{7}$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{- 7 (- \frac{x \cdot 3}{7} + \frac{5}{7}) + 5}{3}$

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

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{- 7 (- \frac{3 x}{7} + \frac{5}{7}) + 5}{3}$

Apply the distributive property.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{- 7 (- \frac{3 x}{7}) - 7 (\frac{5}{7}) + 5}{3}$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{3 x}{7}$ into the numerator.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{- 7 \frac{- 3 x}{7} - 7 (\frac{5}{7}) + 5}{3}$

Factor $7$ out of $- 7$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{7 (- 1) (\frac{- 3 x}{7}) - 7 (\frac{5}{7}) + 5}{3}$

Cancel the common factor.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{7 \cdot (- 1 \frac{- 3 x}{7}) - 7 (\frac{5}{7}) + 5}{3}$

Rewrite the expression.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{- 1 (- 3 x) - 7 (\frac{5}{7}) + 5}{3}$

Multiply $- 3$ by $- 1$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x - 7 (\frac{5}{7}) + 5}{3}$

Cancel the common factor of $7$ .

Factor $7$ out of $- 7$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x + 7 (- 1) (\frac{5}{7}) + 5}{3}$

Cancel the common factor.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x + 7 \cdot (- 1 (\frac{5}{7})) + 5}{3}$

Rewrite the expression.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x - 1 \cdot 5 + 5}{3}$

Multiply $- 1$ by $5$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x - 5 + 5}{3}$

Simplify terms.

Combine the opposite terms in $3 x - 5 + 5$ .

Add $- 5$ and $5$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f^{- 1} (- \frac{3}{7} x + \frac{5}{7}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{7 x}{3} + \frac{5}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = - \frac{3}{7} \cdot (- \frac{7 x}{3} + \frac{5}{3}) + \frac{5}{7}$

Simplify each term.

Apply the distributive property.

$f (- \frac{7 x}{3} + \frac{5}{3}) = - \frac{3}{7} \cdot (- \frac{7 x}{3}) - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{- 3}{7} \cdot (- \frac{7 x}{3}) - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Move the leading negative in $- \frac{7 x}{3}$ into the numerator.

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{- 3}{7} \cdot \frac{- 7 x}{3} - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Factor $3$ out of $- 3$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{3 (- 1)}{7} \cdot \frac{- 7 x}{3} - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Cancel the common factor.

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{3 \cdot - 1}{7} \cdot \frac{- 7 x}{3} - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Rewrite the expression.

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{- 1}{7} \cdot (- 7 x) - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Cancel the common factor of $7$ .

Factor $7$ out of $- 7 x$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{- 1}{7} \cdot (7 (- x)) - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Cancel the common factor.

$f (- \frac{7 x}{3} + \frac{5}{3}) = \frac{- 1}{7} \cdot (7 (- x)) - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Rewrite the expression.

$f (- \frac{7 x}{3} + \frac{5}{3}) = - 1 (- x) - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Multiply $- 1$ by $- 1$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = 1 x - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Multiply $x$ by $1$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = x - \frac{3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{7}$ into the numerator.

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + \frac{- 3}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Factor $3$ out of $- 3$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + \frac{3 (- 1)}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Cancel the common factor.

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + \frac{3 \cdot - 1}{7} \cdot \frac{5}{3} + \frac{5}{7}$

Rewrite the expression.

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + \frac{- 1}{7} \cdot 5 + \frac{5}{7}$

Combine $\frac{- 1}{7}$ and $5$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + \frac{- 1 \cdot 5}{7} + \frac{5}{7}$

Multiply $- 1$ by $5$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + \frac{- 5}{7} + \frac{5}{7}$

Move the negative in front of the fraction.

$f (- \frac{7 x}{3} + \frac{5}{3}) = x - \frac{5}{7} + \frac{5}{7}$

Combine the opposite terms in $x - \frac{5}{7} + \frac{5}{7}$ .

Add $- \frac{5}{7}$ and $\frac{5}{7}$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = x + 0$

Add $x$ and $0$ .

$f (- \frac{7 x}{3} + \frac{5}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{7 x}{3} + \frac{5}{3}$ is the inverse of $f (x) = - \frac{3}{7} x + \frac{5}{7}$ .

$f^{- 1} (x) = - \frac{7 x}{3} + \frac{5}{3}$

Find the Inverse y=-3/7x+5/7

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