Number of solutions of a system of equations

I'm preparing for the ACTM State contest, and I stumbled across this question.

Determine the number of solutions in the system of equations:

\begin{align*} -x+6y-3z &= -8\\ x-2y+2z &= 3\\ 3x+2y+4z &= -6. \end{align*}

I know how to solve a system of equations, but I was wondering if anyone knows a shortcut to this problem. Is there any way of telling how many solutions a system has without actually solving it?

• Normally, a system of equations has one solution. If they ask you how many solutions it has, you should suspect that it is not one of the normal cases. – Fabio Somenzi Apr 15 '17 at 14:10
• What do you mean by "normally"? Consider the system $$x = 0 \\ x = 1$$, which definitely has no solution at all...so the term "normally" in this context doesn't make sense in my opinion... – ComplexFlo Apr 16 '17 at 11:04

adding the first two equations we obtain: $$4y-z=-5$$ multiplying the first equation by $3$ and adding to the third we have $$20y-5z=-30$$ dividing by $5$ $$4y-z=-6$$ thus our System has no Solutions.