The equation given $18 = 3x + 2y + 1/2z$ where $x, y, z$ are positive integers and $20 = x+y+z$.

Now, not really knowing what to do, I multiplied through by $2$ to make it a linear Diophantine equation with three unknowns, this gives $36 = 6x + 4y + z$. Solving this using a technique I found on this site I was able to obtain a solution to the original equation: $18 = 3(36) + 2(-36) + (-36)1/2$.

As I said, that is a solution, but not within the constraints I want. $x, y, z$ are not all positive integers and they also don't sum up to $20$. (The correct solution is $x = 2$ and $y = 2$ and $z = 16$, found with Wolfram Alpha.)

Now, assuming I know how to solve linear Diophantine equations with three unknowns, how would I take the constraint into account that $x,y,z$ need to sum up to a certain positive integer? What would be the approach here to find the correct solution?

  • 1
    $\begingroup$ Solving both, you get $5x + 3y = 16$ which may make it easier to see only one possible values for $x, y$ and then plug in $x + y + z = 20$ to find $z$. $\endgroup$
    – Math Lover
    Dec 12, 2020 at 16:12

1 Answer 1


$$ \begin{cases} 6 x+4 y+z=36\\ x+y+z=20\\ \end{cases} $$ Subtract the two equations $$5x+3y=16\tag{1}$$ which is satified for positive integers only by $x=2,y=2$.

substitute in the second equation $z=20-x-y$ and get $z=16$, so the unique solution is $x=2,y=2,z=16$.


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