My approach
So here I formed 3 equations with 3 variables.
$$\left\{\begin{array}{c} x^2 + \sqrt{3}xy + y^2 = 25 \\ x^2 + xz + z^2 = 16 \\ y^2 + z^2 = 9 \end{array}\right.$$
Solving these equation seemed impossible so I tried hit and trial and tried inserting integral values for like $z= 3$ then $y=0$ which gave $x=5$ but contradicted the 3rd equation so is there any simpler way ?
It would be great help if someone provides a link for mathjax