Solve a system of equation 
Solve this system of equations.
  $$\begin{cases}
x^{2} + y^{2} = 4 \\ 
z^{2} + t^{2} = 9 \\
xt + yz = 6
\end{cases}$$

My attempts: 


*

*$(x^{2}+y^{2})(z^{2} + t^{2})=36 \xrightarrow{\text{by applying Lagrange formula}} (xt + zy)^{2} + (zx -ty)^{2}=36 \xrightarrow{xt + yz = 6}  36 + (zx -ty)^{2} =36\Rightarrow (zx -ty)^{2} = 0 \Rightarrow zx = ty \Rightarrow z = \frac{ty}{x}$

*$xt + yz = 6 \xrightarrow{z = \frac{ty}{x}}  xt + \frac{ty^{2}}{x} = 6 \Rightarrow x^{2}t + y^{2}t = 6x \Rightarrow t(x^{2} + y^{2})=6x \xrightarrow{x^{2} + y^{2} = 4} t=\frac{3}{2}x$

*$z= \frac{ty}{x} \xrightarrow{t=\frac{3}{2}x} z = \frac{3}{2}y$ 
Can we go further and determine the exact value of variables?

You have $4$ variable and $3$ equation. So one of the variables is free and you can assign some value to it, for example $y=1$, so you have $z=\frac{3}{2}$ , $x=\sqrt{3}$ , $t=\frac{3}{2}\sqrt{3}$.  
