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?

  • $\begingroup$ 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. $\endgroup$ Commented Apr 15, 2017 at 14:10
  • $\begingroup$ 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... $\endgroup$
    – ComplexF
    Commented Apr 16, 2017 at 11:04

1 Answer 1


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.

  • $\begingroup$ I don't think this answers the question asked. They said they know how to solve and seemed wanted a general result. Not a specialized ad hoc method. $\endgroup$ Commented Apr 16, 2017 at 11:08
  • $\begingroup$ this isn't an ad hoc method $\endgroup$ Commented Apr 16, 2017 at 11:24
  • $\begingroup$ It isn't? [extra filler characters] $\endgroup$ Commented Apr 16, 2017 at 11:27
  • $\begingroup$ this answers exactly the given question $\endgroup$ Commented Apr 16, 2017 at 11:29
  • $\begingroup$ Disinformation... guess that makes your question correct doesn't it. Have a wonderful Easter. $\endgroup$ Commented Apr 16, 2017 at 11:32

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