To find the inverse of the function $f(x) = -\frac{2}{3}x + 10$ in slope-intercept form, follow these steps:

Step 1: Replace $f(x)$ with $y$

$y = -\frac{2}{3}x + 10$

Step 2: Swap $x$ and $y$

$x = -\frac{2}{3}y + 10$

Step 3: Solve for $y$ (the new function will be the inverse)

1. Subtract 10 from both sides: $x - 10 = -\frac{2}{3}y$

2. Multiply both sides by the reciprocal of $-\frac{2}{3}$, which is $-\frac{3}{2}$: $y = -\frac{3}{2}(x - 10)$

3. Distribute $-\frac{3}{2}$ across the expression: $y = -\frac{3}{2}x + 15$

Step 4: Write the inverse function

$f^{-1}(x) = -\frac{3}{2}x + 15$

This is the inverse function in slope-intercept form.

Would you like further details or have any questions? Here are some related problems you might consider:

1. Find the inverse of $f(x) = \frac{4}{5}x - 7$.
2. Determine the slope and y-intercept of $f^{-1}(x) = -\frac{3}{2}x + 15$.
3. Verify if $f(f^{-1}(x)) = x$ using the original and inverse functions.
4. Solve for $f^{-1}(x)$ if $f(x) = -\frac{2}{3}x + c$, where $c$ is a constant.
5. Graph the original function and its inverse on the same coordinate plane.

Tip: The graph of a function and its inverse are always reflections of each other across the line $y = x$.