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.
$\endgroup$2 Answers
$\begingroup$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:
$$x=\frac{1}{21}(11+4\sqrt{7})$$
$$y=\frac{2}{7}(11+4\sqrt{7})$$
$\endgroup$ 9 $\begingroup$Squaring 1st equation you have $3x(x+y+1)^2=4(x+y)^2$, squaring second you have $7y(x+y-1)^2=32(x+y)^2$. Put $s=x+y,y=s-x$ and we get $3s^2x+6sx+3x-4s^2=0,7s^3-7s^2x+14sx-46s^2+7s=0$. The first of these gives $x=\frac{4s^2}{3s^2+6s+3}$.
Substituting in the second we get $$s(21s^4-124s^3-178s^2-124s+21)=0$$ or $$s(3s^2-22s+3)(7s^2+10s+7)=0$$ It is clear from the original equations that $s\ne0$ and $7s^2+10s+7=0$ has no real solutions, so we have $s=\frac{1}{3}(11\pm4\sqrt7)$.
Using the equation for $x$ we get $x=\frac{1}{21}(11+4\sqrt7),y=\frac{2}{7}(11+4\sqrt7)$ or $x=\frac{1}{21}(11-4\sqrt7),y=\frac{22}{7}(11-\frac{8}{\sqrt7})$. But checking with the original equations the second solution fails (because $s-1$ is negative).
$\endgroup$ 5
