<s>[INST] Solving 2-degree equations in 3 variables. We are given 3 equations:
$x^2+\sqrt3 xy + y^2 = 25$
$y^2 + z^2 = 9$
$x^2 +xz+ z^2 = 16$.
$x,y,z$ are positive real numbers.
Then we have to find value of $xy + 2yz + \sqrt3 xz$.
 [/INST] Based on system of equations being sides of a right angled triangle and a point P
inside the triangle such that -
$\angle BPC = 90^0, \angle APC = 150^0, \angle APB = 120^0 $
and, $AP = x, CP = y, BP = z$
We know area of a triangle is $\frac{1}{2} \times$ length of side 1 $\times$ length of side 2 $\times \sin \theta$
where $\theta$ is the angle between side 1 and side 2.
Now sum of area, $\triangle APC + \triangle BPC + \triangle APB = \triangle XYZ$
$\frac{1}{2}(xy\sin150^0 + yz\sin90^0 + xz\sin 120^0) = \frac{1}{2} \times 3 \times 4$
$xy \times \frac{1}{2} + yz + xz \times \frac{\sqrt3}{2} = 12$
$xy + 2yz + \sqrt3 xz = 24$
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