i'm back.

so i now understand all of the above. and my understanding is that

F = (-201.5y + 111.5) / (75 - 201.5y)

if we give this thing a y value, it should give us the corresponding F value which makes it = 0 (because that was our starting premise). the reason it should = 0 is because we are looking to find what the worst hand in our 5b shoving range can be based on certain amounts of fold equity. (F is villain's fold %, y is the equity of the worst hand in our shove range).

so i went ahead and started messing aboot with some different y values. in this particular case, i chose 0.35.

F = (-201.5 * 0.35) + 111.5 / 75 - (201.5 * 0.35)
F = -70.525 + 111.5 / 75 - 70.525
F = 40.975 / 4.475
F = 9.156

so thus i assumed that if y = 0.35, F = 0.915 (because F has to be between 0.00 and 1.00, i assume i should take that to mean 9% of the time, or 0.09)

so i then plugged both back into the original equation, to see if they equalled zero.

0 = 36.5F + (1-F)(-90y) + (1 - F - y + Fy)(111.5)
0 = 33.21 + 0.909 * 31.5 + 0.59 (111.5)
0 = 3.321 - 28.633 + 65.875
0 = 40.563

0 obviously does not equal 40.563. and thus i have either fucked up the calculations. or completely missed the point of this equation.

so my questions for you maths people out there are:
- is there a mistake in my working out?
- am i wrong in assuming if i solve the first one, the F and y variables should be able to be plugged into the original equation, and give an outcome of 0?