Find the Inverse 3x-2=5y+7

$3 x - 2 = 5 y + 7$

Step 1

Rewrite the equation as $5 y + 7 = 3 x - 2$ .

$5 y + 7 = 3 x - 2$

Step 2

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

Subtract $7$ from both sides of the equation.

$5 y = 3 x - 2 - 7$

Subtract $7$ from $- 2$ .

$5 y = 3 x - 9$

Step 3

Divide each term in $5 y = 3 x - 9$ by $5$ and simplify.

Divide each term in $5 y = 3 x - 9$ by $5$ .

$\frac{5 y}{5} = \frac{3 x}{5} + \frac{- 9}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{3 x}{5} + \frac{- 9}{5}$

Divide $y$ by $1$ .

$y = \frac{3 x}{5} + \frac{- 9}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{3 x}{5} - \frac{9}{5}$

Step 4

Interchange the variables.

$x = \frac{3 y}{5} - \frac{9}{5}$

Step 5

Solve for $y$ .

Rewrite the equation as $\frac{3 y}{5} - \frac{9}{5} = x$ .

$\frac{3 y}{5} - \frac{9}{5} = x$

Add $\frac{9}{5}$ to both sides of the equation.

$\frac{3 y}{5} = x + \frac{9}{5}$

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

$\frac{5}{3} \cdot \frac{3 y}{5} = \frac{5}{3} (x + \frac{9}{5})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{5}{3} \cdot \frac{3 y}{5}$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5}{3} \cdot \frac{3 y}{5} = \frac{5}{3} (x + \frac{9}{5})$

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{5}{3} (x + \frac{9}{5})$

Cancel the common factor of $3$ .

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{5}{3} (x + \frac{9}{5})$

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{5}{3} (x + \frac{9}{5})$

Rewrite the expression.

$y = \frac{5}{3} (x + \frac{9}{5})$

Simplify the right side.

Simplify $\frac{5}{3} (x + \frac{9}{5})$ .

Apply the distributive property.

$y = \frac{5}{3} x + \frac{5}{3} \cdot \frac{9}{5}$

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

$y = \frac{5 x}{3} + \frac{5}{3} \cdot \frac{9}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y = \frac{5 x}{3} + \frac{5}{3} \cdot \frac{9}{5}$

Rewrite the expression.

$y = \frac{5 x}{3} + \frac{1}{3} \cdot 9$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

Step 7

Verify if $f^{- 1} (x) = \frac{5 x}{3} + 3$ is the inverse of $f (x) = \frac{3 x}{5} - \frac{9}{5}$ .

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 $f^{- 1} (\frac{3 x}{5} - \frac{9}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{5 (\frac{3 x}{5} - \frac{9}{5})}{3} + 3$

Simplify each term.

Simplify the numerator.

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{5 (\frac{3 x - 9}{5})}{3} + 3$

Factor $3$ out of $3 x - 9$ .

Factor $3$ out of $3 x$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{5 (\frac{3 (x) - 9}{5})}{3} + 3$

Factor $3$ out of $- 9$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{5 (\frac{3 x + 3 \cdot - 3}{5})}{3} + 3$

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{5 (\frac{3 (x - 3)}{5})}{3} + 3$

Combine $5$ and $\frac{3 (x - 3)}{5}$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{\frac{5 (3 (x - 3))}{5}}{3} + 3$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{\frac{15 (x - 3)}{5}}{3} + 3$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{15 (x - 3)}{5}$ by cancelling the common factors.

Factor $5$ out of $15 (x - 3)$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{\frac{5 (3 (x - 3))}{5}}{3} + 3$

Factor $5$ out of $5$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{\frac{5 (3 (x - 3))}{5 (1)}}{3} + 3$

Cancel the common factor.

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{\frac{5 (3 (x - 3))}{5 \cdot 1}}{3} + 3$

Rewrite the expression.

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{\frac{3 (x - 3)}{1}}{3} + 3$

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

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{3 (x - 3)}{3} + 3$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = \frac{3 (x - 3)}{3} + 3$

Divide $x - 3$ by $1$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = x - 3 + 3$

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

Add $- 3$ and $3$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{3 x}{5} - \frac{9}{5}) = x$

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

Set up the composite result function.

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

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

$f (\frac{5 x}{3} + 3) = \frac{3 (\frac{5 x}{3} + 3)}{5} - \frac{9}{5}$

Combine the numerators over the common denominator.

$f (\frac{5 x}{3} + 3) = \frac{3 (\frac{5 x}{3} + 3) - 9}{5}$

Simplify each term.

Apply the distributive property.

$f (\frac{5 x}{3} + 3) = \frac{3 (\frac{5 x}{3}) + 3 \cdot 3 - 9}{5}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{5 x}{3} + 3) = \frac{3 (\frac{5 x}{3}) + 3 \cdot 3 - 9}{5}$

Rewrite the expression.

$f (\frac{5 x}{3} + 3) = \frac{5 x + 3 \cdot 3 - 9}{5}$

Multiply $3$ by $3$ .

$f (\frac{5 x}{3} + 3) = \frac{5 x + 9 - 9}{5}$

Simplify terms.

Combine the opposite terms in $5 x + 9 - 9$ .

Subtract $9$ from $9$ .

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

Add $5 x$ and $0$ .

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{3} + 3$ is the inverse of $f (x) = \frac{3 x}{5} - \frac{9}{5}$ .

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

Find the Inverse 3x-2=5y+7

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