Given 4f(x+5)+2g(x-3)=3x+10 and 3f(x+5)-g(x-3)=x, what are the functions f and g?

We'll note 4f(x+5) + 2g(x-3) = 3x+10
(1)

3f(x+5) - g(x-3) = x
(2)

We'll multiply 3f(x+5) - g(x-3) = x by
2:

6f(x+5) - 2g(x-3) = 2x
(3)

We'll add (3) +
(1):

6f(x+5) - 2g(x-3) + 4f(x+5) + 2g(x-3) = 2x + 3x +
10

We'll eliminate like
terms:

10f(x+5) = 5x +
10

We'll divide by 5:

2f(x+5)
= x + 2

f(x+5) = x/2 + 1

But
f(x) = ax + b

f(x+5) = a(x+5) +
b

a(x+5) + b = x/2 + 1

We'l
remove the brackets:

ax + 5a + b =  x/2 +
1

Comparing, we'll get:

a =
1/2

5a + b = 1 => 5/2 + b = 1 => b = 1 - 5/2
=> b = -3/2

f(x) = x/2 -
3/2

We'll replace f(x+5) in
(2):

3f(x+5) - g(x-3) = x

3x/2
+ 3 - g(x-3) = x

g(x-3) = 3x/2 + 3 -
x

g(x-3) =  x/2 + 3

But g(x) =
cx + d

g(x-3) = cx - 3c + d

cx
- 3c + d =  x/2 + 3

Comparing, we'll
get:

c = 1/2

-3c+d = 3
=> -3/2 + d = 3 => d = 3 + 3/2 => d =
9/2

g(x) = x/2 +
9/2

The requested functions f and g, that
respect the given conditions, are: f(x) = x/2 - 3/2 and g(x) = x/2 +
9/2.