Solve for x in terms of t:

x-1 x-1-6t 22
------ + -------- = ----
x-1-t x-1-7t 7

Whenever tâ‰ 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**.