# 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?

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x-2 = 2f(x-1) + 3g(x+1).............(1)

-6x + 4=4f(x-1) - 2g(x+1)........(2)

==> -2*(1) + (2) =

==> -2x + 4 = -4f(x-1) + -6g(x+1)

==> -6x + 4 = 4f(x-1) - 2g(x+1)

==> -8x + 8 = -8g(x+1)

Divide by -8:

==> g(x+1) = x-1

==> substitute g(x+1) in (1)

2f(x-1) + 3g(x+1) = x-2

2f(x-1) + 3(x-1) = x-2

2f(x-1) + 3x-3 = x-2

2f(x-1) = x-2 -3x + 3

2f(x-1) = -2x + 1

f(x-1) = -x + 1/2

==> f(x) = -x - 1 + 1/2

**==> f(x) = -x - 1/2**

**==> g(x) = x-2 **

We'll write the linear functions f(x) and g(x):

f(x) = ax + b

g(x) = cx + d

Now, we'll note f(x-1) = u and g(x+1) = v

We'll re-write the system of the constraints from enunciation:

2u + 3v = x - 2 (1)

4u - 2v = -6x+4 (2)

We'll solve the system using elimination method. For this reason, we'll multiply (1) by 2 and (2) by 3:

4u + 6v = 2x- 4 (3)

12u - 6v = -18x+12 (2)

We'll add (3) + (2):

4u + 6v + 12u - 6v = 2x- 4 - 18x + 12

We'll combine and eliminate like terms:

16u = -16x + 8

We'll divide by 16:

**u = -x + 1/2**

4u - 2v = -6x+4

4(-x + 1/2) - 2v = -6x+4

We'll remove the brackets:

-4x + 2 - 2v = -6x + 4

-2v = -6x + 4x - 2 + 4

-2v = -2x + 2

**v = x - 1**

But f(x-1) = u

f(x-1) = -x + 1/2

f(x-1) = a(x-1) + b

-x + 1/2 = ax - a + b

a = -1

b - a = 1/2

b + 1 = 1/2

b = -1/2

**f(x) = -x - 1/2**

g(x+1) = x - 1

g(x+1) = c(x+1) + d

x - 1 = cx + c + d

c = 1

c + d = -1

1 + d = -1

d = -2

**g(x) = x - 2**

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.