Give the solution expressed as a Particular+Homogeneous

Question

I was given the set of linear equations x1 + x2 = 3

x1 + 2x2 = 4

2x1 + x2 = 5

I solved the equation and got x1=2 and x2=1

However, I am not sure how to express this as a Particular + Homogeneous solution.

Answer 1

You have to solve the system

$$x_1+x_2=0$$

$$x_1+2x_2=0$$

$$2x_1+x_2=0$$

If we subtract the first two equations, we immediately get $x_2=0$, which implies $x_1=0$. So, the homogenous system has only the trivial solution.

This means, that the original system has the unique solution you mentioned.

Formally , you could write this as a sum of your particular solution and the general solution of the homogenous system (which is identically $0$).