I need to find all integer solutions of the following equation:

$$2x + 3y + 4z = 5$$

So far I have already found, through a system of 2 equations in 2 variables, that one particular solution is $x = 15, y = -15,$ and $z = 5$.

There is a formula for finding all the integer solutions from two known solutions for two variables, I'm wondering if there is one for three? I imagine it would have to do with the gcd's of $2, 3$ and $4$.I just don't know quite what it would look like.

Thanks in advance!


If you want the set of all solutions of this problem, then all you need to do, is the following:

Note that $2x+3y+4z=5 \iff 4(z+2x) + 3(y-2x) = 5$. Now, the general solution for an equation of this form is that $(z+2x,y-2x) = (-3n-1,4n+3)$, using the ordinary technique you have for two variables.

Hence, the final solution is $(x,y,z) = (k, 4n+2k+3,-3n-2k-1)$, where $n,k$ can vary among the integers.

For example, $n=13,k=-12$ gives $x = -12, y= 31, z = -16$, and $2x+3y+4z = 5$.

What helped here is the technique of reducing variables. Hence, we get the result.

  • $\begingroup$ That does make sense. But is there a way to do it using my technique: the system of 2 equations in 2 variables? That's the way I was introduced to. $\endgroup$ – Francesca Oct 6 '16 at 3:18
  • $\begingroup$ I think I did the same thing, because I performed the reduction using a small trick. $\endgroup$ – астон вілла олоф мэллбэрг Oct 6 '16 at 3:19

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.