Diophantine system of two equations with four variables

Question

Find all integer solutions for the system:

$$\left\{\begin{array}{rcl}xy + vw &=& 5 \\ xv - yw &=& 6\end{array}\right.$$

It's supposed to be solvable by 9-graders...

Answer 1

By Lagrange's identity we have: $$ (xy+vw)^2+(xv-yw)^2 = (x^2+w^2)(v^2+y^2) $$ but since $5^2+6^2 = 61$ is a prime number any integer solution of the initial system of equations must satisfy: $$\left\{\begin{array}{rcl} x^2+w^2&=&1\\ v^2+y^2&=&61\end{array}\right.\quad\text{or}\quad \left\{\begin{array}{rcl} x^2+w^2&=&61\\ v^2+y^2&=&1,\end{array}\right.$$ so one of the variables is zero.

Answer 2

$$ (x + iw)(v+iy) = 6 + 5i$$

This implies (taking modulus of the complex numbers)

$$ (x^2 + w^2)(v^2 + y^2) = 61$$

Since $61$ is prime...

[Heavyweight: this is just factorizing $6+5i$ in the Gaussian integers]