Solving complicated system of equations:

Question

I'm trying to solve this system of equations, and was wondering if there is any possible algebraic manipulation I can do to solve this question.

Here are the equations:

$x+y+z=338$

$xy+yz+zx=335$

EDIT: Sorry, solve for such that x,y,z are positive integers

Thanks.

Answer 1

I used substation and solved the first equation for x then plugged into the second one. I tried to use completing the square but I don't think it quite works out. When you graph it, its an ellipse.

Answer 2

Since you have three variables but only two equations, you must keep one variable as a parameter.

On the other side, if we keep $z$ as a free parameter, also notice that $x$ and $y$ play the same role.

So, from the first equation $y=338-x-z$. Plug in the second to get $$-\left(z^2-338 z+335\right)+ (338-z)x-x^2=0$$ Solve the quadratic in $x$ to make the solutions to be $$x=\frac{1}{2} \left(-\sqrt{-3 z^2+676 z+112904}-z+338\right)\qquad y=\frac{1}{2} \left(\sqrt{-3 z^2+676 z+112904}-z+338\right)$$ which, in the real domain will exist if $$\frac{2}{3} \left(169-7 \sqrt{2311}\right)\leq z \leq \frac{2}{3} \left(169+7 \sqrt{2311}\right)$$