How to solve the "four" variables problem Given x, y, z, and w are real numbers which satisfy these three equation:
$$x^{2} + 5z^{2} = 10$$
$$yz - xw = 5$$
$$xy + 5zw = \sqrt{105}$$
Find the value of $y^{2} + 5w^{2}$
Can anyone give a hint? I don't even know where to start

Method$\#1:$
Solve the two simultaneous equations in $x,z$
$$z=\dfrac{aw+5y}{y^2+5w^2},x=\dfrac{ay-25w}{y^2+5w^2}$$
Replace the values of $z,w$ in $$10=x^2+5z^2$$
$$10=\dfrac{a^2y^2+625w^2-50ayw+5(a^2w^2+25y^2+10ayw)}{(y^2+5w^2)^2}$$
$$=\dfrac{(a^2+125)y^2+(625+5a^2)w^2}{(y^2+5w^2)^2}=\dfrac{a^2+125}{y^2+5w^2}$$
where $a^2=105$
See also : Brahmagupta-Fibonacci Identity
Method$\#2:$
WLOG $x=\sqrt{10}\cos t, z=\sqrt2\sin t$
$$yz - xw = 5\implies y(\sqrt2\sin t) - (\sqrt{10}\cos t)w = 5$$
$$xy + 5zw = \sqrt{105}\implies 5(\sqrt2\sin t)w+ (\sqrt{10}\cos t)y  = \sqrt{105} $$
Solve for $\cos t,\sin t$ and use $\cos^2t+\sin^2t=1$ to eliminate $t$
