Question

How do I solve this manually?

http://pasteboard.co/7F49Diqrd.jpg

Answer 1

You can set a system of equations or use matrices.

Answer 2

could you please elaborate? I'm not familiar with those methods.
Thank you, in advance!

Answer 3

Try to find the simplest equation with few unknowns.
Then sub in gradually into the others until you get your result

A + C = 0 means A = -C
Replace all A in other eqns with -C like so:

-4A+B-8C + D = 1 --> 4C + B - 8C + D = 1 so -4C + B + D = 1 (ii)
3A+16C-8D = -29 becomes 13C - 8D = -29 (iii)
-12A+3B+16D=5 becomes 12C + 3B + 16D = 5 (iv)

Now you have 3 eqns.
2 of these have C, B and D (ii) and (iv)
One has C and D. (iii)
You could again use the simplest equation with fewest unknowns (iii) to sub into the rest, but it might be easier to focus on (ii) and (iv) and eliminate one of the variables to make it the same as (iii)

-4C + B + D = 1 (ii)
12C + 3B + 16D = 5 (iv)

To eliminate, we must remove the variable B to match (iii) which only has C and D
(ii) x 3 --> -12C + 3B + 3D = 3 (v)

Then do (v) - (iv):

-12C + 3B + 3D = 3 (v) -
12C + 3B + 16D = 5 (iv)
-24C - 13D = -2 (vi)

Ok now we have an equation (vi) with 2 unknowns in C and D just like eqn (iii).
So we will solve simultaneously by elimination again to find C and D

13C - 8D = -29 (iii)
-24C - 13D = -2 (vi)
This one is not as nice. The LCM of 13 and 24 is 312.

Anyhow (iii) x 24 and (vi) x 13 gives
312C - 192D = 232 +
-312C - 169D = -26
361D = 206

I made have made a mistake somewhere but you get the idea now.