How can I solve this hard system of equations? Solve the system below

\begin{align}
&\sqrt {3x} \left( 1+\frac {1}{x+y}  \right) =2\\
&\sqrt {7y} \left( 1-\frac{1}{x+y}  \right) =4\sqrt{2}
\end{align}

Frankly I am disappointed, because I spent around 2 hours in solving this equation, but, finally I didn't do it so, I hope you can help me in approaching this problem.

Square both equations. Then let $u=\frac{1}{x+y}$. Note then $x+y=\frac{1}{u}$
$$3x(1+\frac{1}{x+y})^2=4$$
$$7y(1-\frac{1}{x+y})^2=32$$
So.
$$x(1+u)^2=\frac{4}{3}$$
$$y(1-u)^2=\frac{32}{7}$$
And for $u \neq 1$
$$x=\frac{4}{3}(1+u)^{-2}$$
$$y=\frac{32}{7}(1-u)^{-2}$$
Adding both equations we have:
$$\frac{1}{u}=\frac{4}{3(1+u)^2}+\frac{32}{7(1-u)^2}$$
$$\frac{1}{u}=\frac{28(1-u)^2+96(1+u)^2}{21(1-u)^2(1+u)^2}$$
Cross multiplication, and moving terms to one side yields:
$$28u(1-u)^2+96u(1+u)^2-21(1-u)^2(1+u)^2=0$$
$$-21u^4+124u^3+178u^2+124u-21=0$$
This factors:
$$-(3-22u+3u^2)(7+10u+7u^2)=0$$
So:
$$u=\frac{11 \pm 4\sqrt{7}}{3}$$
$$x+y=\frac{3}{11 \pm 4\sqrt{7}}$$
Back substitution yields:
$$\sqrt{3x}(\frac{14 \pm 4\sqrt{7}}{3})=2$$
$$x=\frac{12}{(14 \pm 4\sqrt{7})^2}$$
$$y=\frac{3}{11 \pm 4\sqrt{7}}-\frac{12}{(14 \pm 4\sqrt{7})^2}$$
We introduced an extraneous solution when squaring so checking the possible solutions, the final solution is:
$$x=\frac{12}{(14-4\sqrt{7})^2}$$
$$y=\frac{3}{11-4\sqrt{7}}-\frac{12}{(14- 4\sqrt{7})^2}$$
If you wish, from the second equation, you can express $y$ as:
$$y=\frac{288}{7(4\sqrt{7}-8)^2}$$
In fact through the method of multiplying by conjugates we may get:
$$x=\frac{1}{21}(11+4\sqrt{7})$$
$$y=\frac{2}{7}(11+4\sqrt{7})$$
