Question

Solve these equations $\begin{array}{c}x\sqrt{x}+y\sqrt{y}=183\\ x\sqrt{y}+y\sqrt{x}=182\end{array}$

Answer 1

$\begin{array}{c}u-t=\sqrt{{\{u+t\}}^2-4ut}\\ =\sqrt{{\left\{9\right\}}^{2-}4\times \frac{182}{9}}\\ =\sqrt{81-\frac{728}{9}}\\ =\sqrt{\frac{729-728}{9}}\\ =\sqrt{\frac{1}{9}}\\ u-t=\frac{1}{3}\\ u+t=9\\ \_\_\_\_\_\_\_\_\_\_\\ 2u=\frac{1}{3}+9\\ 2u=\frac{28}{3}\\ u=\frac{14}{3}\\ alsou+t=9\\ t=9-u\\ =9-\frac{14}{3}\\ =\frac{27-14}{3}\\ t=\frac{13}{3}\\ vy=uory=u^2=\{\frac{14}{3}{\}}^2=\frac{196}{9}\\ vx=torx=t^2=\{\frac{13}{3}{\}}^2=\frac{196}{9}\\ x=\frac{169}{9},y=\frac{196}{9}\end{array}$