1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Fabio Somenzi Apr 15 '17 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... $\endgroup$ – ComplexFlo Apr 16 '17 at 11:04
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – marshal craft Apr 16 '17 at 11:08
  • $\begingroup$ this isn't an ad hoc method $\endgroup$ – Dr. Sonnhard Graubner Apr 16 '17 at 11:24
  • $\begingroup$ It isn't? [extra filler characters] $\endgroup$ – marshal craft Apr 16 '17 at 11:27
  • $\begingroup$ this answers exactly the given question $\endgroup$ – Dr. Sonnhard Graubner Apr 16 '17 at 11:29
  • $\begingroup$ Disinformation... guess that makes your question correct doesn't it. Have a wonderful Easter. $\endgroup$ – marshal craft Apr 16 '17 at 11:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.