A system of linear equations with solution $x=20$ and $y=20$ I am trying to come up with a system with solution $x=20$ and $y=20$. This is what I have now:

$$\begin{array}{|l} 3(x-3y)-(2x-3y)=-100 \\ 4(x-3y)+2(2x-3y)=-200\end{array}$$
The expected solution is to define $a=x-3y$ and $b=2x-3y$ and solve in this way. Do you think it's okay right now or the solution's obvious? How can I make it more complicated?
Your system is fine.

If you want to make it more complicated make it nonlinear. 

For example $$ (2x-y)^2+(x+y)^2=2000, 3(2x-y)+(x+y)^2=1660$$
Simplifying your system we get
$$x-6y=-100$$
$$4x-9y=-100$$
Multiplying the first equation by $-4$ and adding to the second we get
$$150y=300$$ so $y=20$ as you stated.