Solve non-linear system of four variable

Question

I have the following nonlinear system:

\begin{align} 2x-2a-b&=0\\ 2y-a+b&=0\\ a(-2x-2y-1)&=0\\ b(-x+y+4)&=0\\ a, b &\geq 0 \end{align}

We can see the solutions here.

My question is - how can I solve this system as simply as possible "on paper"?

Answer 1

Hint

Sum two first equations and get:

$$2x+2y=3a \quad (1)$$

Plug it in the third equation:

$$a(-3a-1)=0 \rightarrow a=0 \quad \text{or} \quad a=-1/3 \quad (2)$$

Subtract two first equations and get:

$$x-y=\frac{a+2b}{2} \quad (3)$$

Plug it in the fourth equation:

$$b\left(-\frac{a+2b}{2}+4\right)=0 \rightarrow b=0 \quad \text{or} \quad a+2b=8 \quad (4)$$

Using $(2)$ and $(4)$ you can find $a$ and $b$. Then back to $(1)$ and $(3)$ and find $x$ and $y$.

Can you finish?