0
$\begingroup$

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?

$\endgroup$
1
  • 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

3
$\begingroup$

$$ \begin{cases} 6 x+4 y+z=36\\ x+y+z=20\\ \end{cases} $$ 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 $x=2,y=2,z=16$.

$\endgroup$

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.