# Linear functionsCalculate 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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### 1 Answer

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