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**A Beautiful Mind** This seems like it requires a lot of intuition because I'm always having trouble figuring the right answers and get different answers...

Like here:

$\displaystyle 3x-5y+5z = 1$

$\displaystyle -5x-2y+3z = 0$

$\displaystyle 7x-y+3z = 0$

My work:

$\displaystyle -5x-2y+3z = 0$

$\displaystyle -7x+y-3z = 0$

$\displaystyle -12x-y = 0$

$\displaystyle 7x +y+3z = 0$

$\displaystyle

-5x+3z = 0$

$\displaystyle 5x+2y-3z=0$

$\displaystyle 2y = 0$

Inconsistent. And this isn't how you do it and I don't know why what I'm doing is wrong.