A friend of Mine gave me a system of two equations and asked me to solve them $\rightarrow$
$\sqrt{x}+y=11~~ ...1$ $\sqrt{y}+x=7~~ ...2$
I tried to solve them manually and got this horrendously complicated fourth degree equation $\rightarrow$
$\begin{align*} y &= (7-x)^2 ~...\mbox{(from 2)} \\ y &= 49 - 14 x + x^2 \\ \implies 11&= \sqrt{x}+ 49 - 14 x + x^2 ...(\mbox{from 1)}\\ \implies~~ 0&=x^4-28x^3+272x^2-1065x+1444 \end{align*}$
Solving this wasn't exactly my piece of cake but I could tell that one of Solutions would have been 9 and 4
But my friend kept asking for a formal solution.
I tried plotting the equations and here's what I got $\rightarrow$
So the equations had two pairs of solutions (real ones).
Maybe, Just maybe I think these could be solved using approximations.
So How do i solve them using a formal method (Calculus,Algebra,Real Analysis...)
P.S. I'm In high-school.