Fastest way to solve the following system

Question

$−y +z −3t = 0$

$3x +4y +6z −6t = 0$

$x −y +2z −2t = 0$

What is the fastest method? I usually use substitution but it takes time in higher dimensions. I already know that the rank of the matrix is $3$.

Answer 1

Rewriting your system in the form $$x-y+2z-2t=0$$ $$3x+4y+6z-6t=0$$ $$-y+z-3t=0$$ Multiplying the first equation by $-3$ and adding to the second one we get $$7y=0$$ Multiplying the first equation by $-1$ and adding to the third we obtain $$-z-t=0$$ Can you finish?

Answer 2

You can solve $x,y,z$ in terms of $t$.

Subtract $3$ times the third equation from the second one and you get $y=0$

Plug in the first one and you get $z=3t$

Plug in the third equation and you get $x=-4t$

Thus your solution is $$(x,y,z)= t(-4,0,3)$$