What are the linear functions f and g if x-2=2f(x-1)+3g(x+1) and 4f(x-1)-2g(x+1)=-6x+4?

To find the linear functions of f and g if x-2 = 2f(x-1)
+3g(x+1).

Let us assume that x-2 is a function of (x-1) 
and (x+1).

So   2k (x-1) +3l(x+1) =
(x-2).

Equating the coeffficients of x and constanrs on
both sides, we get:

(2k+3l)x = x,
Or

2k+3l =
1........(1).

-2k+3l =
-2.........(2).

(1)+(2) gives: 6l = -1, Or l =
-1/6.

(1)-(2) gives: 4k = 3, Or k =
3/4.

Therefore f(x) = (3x/4) and g(x) =
-(x/6)

Verification: 2f(x-1) + 3g(x+1) = 6/4(x-1) -3/6(x+1)
= x-2.