# 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. Commented Apr 15, 2017 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... Commented Apr 16, 2017 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.