Diophantine equations: $x_1y_1+x_2y_2 = x_3y_3+x_4y_4$

Question

Given 3 diophantine equations:

$$x_1y_1+x_2y_2=x_3y_3+x_4y_4$$

and

$$x_1+x_2 = x_3+x_4$$ and

$$y_1+y_2 = y_3+y_4$$

We're interested in solutions to this system of equations when all variables are positive. I conjecture that any solutions have $x_1=x_3$ or $x_1=x_4$ and $y_1=y_3$ or $y_1=y_4$. Any tips on how to go about proving this? Thanks.

Answer 1

No you are wrong. For any $x_1,x_2,x_3,x_4$ let $y_1=x_3, y_2=x_4,y_3=x_1, y_4=x_2$ and you get a solution with all $x$ distinct and all $y$ distinct.

If you want all $8$ numbers distinct, here is an example:

$(x_1,x_2,x_3,x_4)=(5,1,4,2)$

$(y_1,y_2,y_3,y_4)=(11,9,12,8)$