\(\displaystyle 5x^2+2y^2 \ = \ 10 \ and \ 2x^2+5y^2 = 10, \ \implies \ 5x^2+2y^2 \ = \ 2x^2+5y^2\)
\(\displaystyle This \ implies \ that \ 3x^2 \ = \ 3y^2, \ \implies \ |x| \ = \ |y|, \ hence \ x \ = \ \pm y \ and \ y \ = \ \pm x\)
\(\displaystyle Hence, \ four \ solutions, \ see \ graph.\)
\(\displaystyle Note: \ 3x^2 \ = \ 3y^2, \ x^2 \ = \ y^2, \ x \ \ne y, \ however \ if \ x \ = \ y, \ then \ x^2 \ = \ y^2, \ converse \ is \ not \ true.\)
[attachment=0:9yj9dntd]aaa.jpg[/attachment:9yj9dntd]