<s>[INST] Solve $ x^2+y^2=4, z^2+t^2=9, xt+yz=6 $ in integers 
find answers of this system of equations in integers$$
\left\{ 
\begin{array}{c}
x^2+y^2=4 \\ 
z^2+t^2=9 \\ 
xt+yz=6
\end{array}
\right. 
$$

things I have done: we can observe that $$(x^2+y^2)(z^2+t^2)=(xt+yz)^2+(xz-yt)^2 \rightarrow xz-yt=0$$ 
summing up this with first and second equality $$(x+z)^2+(y-t)^2=13$$
at this stage I used guessing answers. putting like $x= 0,z=3,y=2,t=0$.is there a better way to doing this without guessing and making sure that all answer found?   
 [/INST] Letting $u=(x,y)$ and $v=(z,t)$ if $\theta$ is the angle between these two vectors, then:
$$\cos\theta = \frac{u\cdot v}{|u||v|} = \frac{xz+yt}{\sqrt{x^2+y^2}\sqrt{z^2+t^2}}=1$$
So the vectors point in the same direction, and the only real solutions are when $(z,t)=\frac{3}{2}(x,y)$. Then you can try the integer solutions for $x^2+y^2=4$ to find all solutions.
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