Let me give you TWO methods, in detail:
The equation is y = m x + b
The line passes through (-8,9) and (10,-3) when both of the following equations hold:
9 = -8 m + b
-3 = 10 m + b
Substract those two equations from each other to eliminate b. You obtain: 12 = -18 m, or m = -2/3.
Plug that value of m into either equation to obtain the value of b, namely (using the first equation): 9 = -8 (-2/3) + b, so:
b = 9-16/3 = 11/3
If you prefer, you can also obtain the value of b directly by eliminating m between the two equations (multiply the first by 5 and the second by 4 and add them up). You obtain: 45-12 = 9 b, so b = 11/3, indeed.
SECOND METHOD (more efficient):
At a slightly more advanced level, we may obtain the equation of the line directly by stating that (x+8, y-9) is proportional to (10+8,-3-9). If you know what a determinant is, you may just write that the relevant 2 by 2 determinant is zero. Otherwise, just write that the proportionality means:
(x+8)(-3-9) = (y-9)(10+8)
Just simplify that:
-12 (x+8) = 18 (y-9) or 2(x+8)+3(y-9) = 0
This boils down to 2x+3y=11
You may keep this equation as is or divide by 3 to put it in the form
y = (-2/3)x+(11/3) or y = (11-2x) / 3
I advise you to get familiar with the second method ASAP and, also, to acquire a taste for the nicer form of the equation of a line where the coefficient of y need not be 1. Here: 2x+3y=11.
Last step: Check that the equation obtained holds for both points:
2(-8) + 3(9) = 11
2(10) + 3(-3) = 11
Trimming all the fat, the entire computation merely consists in simplifying a single line, as shown in the link below: