System of equations with special substitution

Question

Solve the following system of equations on $\mathbb R$

\begin{cases} \dfrac{x+y}{1+xy}= \dfrac{1-2y}{2-y}\\[6px] \dfrac{x-y}{1-xy}=\dfrac{ 1-3x}{3-x} \end{cases}

Solution

Setting $ \begin{cases} x=\dfrac{u-1}{u+1} \\[4px] y= \dfrac{v-1}{v+1}\end{cases}$

My question: How did they come up with that idea? How about your solution? :)

Thank you in advance !

Answer 1

The substitution reminds the fact that $$ \frac{e^{2t}-1}{e^{2t}+1}=\frac{e^{t}-e^{-t}}{e^{t}+e^{-t}}=\tanh t $$ plus the fact that $$ \tanh(p\pm q)=\frac{\tanh p\pm\tanh q}{1\pm\tanh p\tanh q} $$ so this is essentially the same as setting $x=\tanh p$ and $y=\tanh q$.