This looks like the system to be solved. If so, why is this equation:

mentioned separately? It's not obvious what this equation is doing.x + y + z = 1,

In general, I would probably write this system up as a matrix (you have a homogeneous system, so no need to think about an augmented system) thus:x = 1 - y - z

-10(1 - y - z) + 30y + 10z = 0

-10 + 10y + 10z + 30y + 10z = 0

-10 + 40y + 20z = 0

40y = -20z + 10

8y = -4x + 2

5(1 - y - z) - 40y + 5z = 0

5 -5y - 5z - 40y + 5z = 0

5 - 45y = 0 ??????

and then start doing elementary row operations on it. You follow?