Question

A line passes through the origin and through points A(−2, b−14) and B(14−b, 72). What is the greatest possible value of b?

Answer 1

Since it passes through the origin, there is no y-intercept
and the equation has the form
y = mx
(−2, b−14) and B(14−b, 72)
slope = (72 - b + 14)/(14-b + 2) = (86-b)/(16-b)
the "critical values of" seem to be 86 and 16
If b > 86, the slope is positive
if b < 16, the slope is positive
for 16 < b < 86 the slope is negative

There doesn't seem to be a greatest value of b.
e.g. let b = 1000
then the slope is (86-1000)/(16-1000) = 457/492 = .9288..
and your equation is y = 457/492 x
as x ---> ∞ , slope ----> 1

Answer 2

the slope is constant

(b - 14) / -2 = 72 / (14 - b) ... solve for b

Answer 3

Go with R_scott's answer.
I was overthinking the problem.

Answer 4

also, there is indeed a y-intercept. It just happens to be zero.