Solve for x in terms of t:
x-1 x-1-6t 22
------ + -------- = ----
x-1-t x-1-7t 7
Whenever t≠0
First, I'll just do some algebra:
(x-1)/(x-1-t) + (x-1-6t)/(x-1-7t) = 22/7
1 + t/(x-1-t) + 1 + t/(x-1-7t) = 22/7
t/(x-1-t) + t/(x-1-7t) = 8/7
7t*(x-1-7t) + 7t*(x-1-t) = 8*(x-1-t)*(x-1-7t)
Now for convenience let y=x-1-4t
7t*(y-3t) + 7t*(y+3t) = 8*(y-3t)*(y+3t)
14yt = 8y^2 - 72t^2
4y^2 - 7yt - 36t^2 = 0
4y^2 - 16yt + 9yt - 36t^2 = 0
4y(y-4t) + 9t(y-4t) = 0
(4y+9t) * (y-4t) = 0
Then y=4t or y=-9t/4.
Then back-substitute to get x=8t+1 or x=7t/4+1.
Pencil and Paper
