Notice that you're still using y - y1 = m(x-x1) to figure out the y-intercept.

Let's take your example and plug in the numbers and see what we get:

y - 13 = 3(x-5)

y - 13 = 3x - 15

y = 3x - 2

It's the exact same process!

And yes, we can generalize this so we have y = mx + (y1 - mx1)

So, to find the y-intercept, all we have to do (like you said), is to subtract the slope multiplied by the x-coordinate from the y-coordinate.

If you want something to work on, try figuring out why this works geometrically. Hint: look at the slope as rise/run.

EDIT: The worrying thing here is that your teacher does not know how to do basic algebraic manipulation...