# Beginner: linear system with parameter – how to solve with gaussian elimination?

$$begin{cases}x-ty = 1\(t-1)x -2y = 1end{cases} x,y,t in mathbb{R}$$

So the corresponding matrix is
$$begin{bmatrix}1 & -t & 1 \t-1 & -2 & 1 end{bmatrix}$$

I know by equaling (I) and (II) and substituting (where one has to divide by $$t-2$$ and $$-1-t$$) that for $$t=2$$ and $$t=-1$$ the system has no (unique) solution.

Now I want to solve the system by gaussian elimination and get a general answer for x,y dependent on t.
Problem: If I want to eliminate an coefficient, I need to multiply (I) by $$t-1$$ where $$tneq 1$$ or (II) by $$t$$ where $$tneq0$$.
But this adds an “extra case” because multiplying by $$0$$ is not allowed and my general solutions will not work only for this specific choice of $$t$$.

Can somebody just tell me, what i need to do in order to achieve general solutions with no special case for t by gaussian elimination ? Or am I just getting something wrong ?