Question

u3:04: unit 3 test
find the inverse of the equation
$y = \frac{2}{3}x - 6$
$\bigcirc\\ y = -\frac{3}{2}x + 6$
$\bigcirc\\ y = \frac{3}{2}x + 9$
$\bigcirc\\ y = \frac{2}{3}x + 6$
$\bigcirc\\ y = \frac{3}{2}x + 6$

Explanation:

Step1: Swap x and y

To find the inverse of a function, we first swap the roles of \( x \) and \( y \) in the equation \( y=\frac{2}{3}x - 6 \). So we get \( x=\frac{2}{3}y - 6 \).

Step2: Solve for y

Now we solve the new equation for \( y \). First, add 6 to both sides of the equation:
\( x + 6=\frac{2}{3}y \)
Then, multiply both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \), to isolate \( y \):
\( y=\frac{3}{2}(x + 6) \)
Expand the right - hand side:
\( y=\frac{3}{2}x+\frac{3}{2}\times6 \)
\( y=\frac{3}{2}x + 9 \)

Answer:

\( y=\frac{3}{2}x + 9 \) (the second option: \( y=\frac{3}{2}x + 9 \))