Uncovering Solutions for x and y in Nonlinear Equations

[Albert1]

$\left\{\begin{matrix}
x+y+xy=11 ----(1)& \\
x^2+xy+y^2=19---(2)&
\end{matrix}\right.
$
find all $x,y\in R$

 

[Greg]

Spoiler

\(\displaystyle x+y+xy=11\quad[1]\)

\(\displaystyle x^2+xy+y^2=19\quad[2]\)

\(\displaystyle \text{Note that }x=0,y=0\text{ and }x=y=0\text{ do not give solutions.}\)

\(\displaystyle [1]+[2]\Rightarrow(x+y)^2+(x+y)-30=0\Rightarrow(x+y-5)(x+y+6)=0\)

\(\displaystyle x+y=-6\Rightarrow xy=17\quad(\text{from }[1])\)

\(\displaystyle x+\frac{17}{x}=-6\Rightarrow x^2+6x+17=0\Rightarrow\text{ no solutions.}\)

\(\displaystyle x+y=5\Rightarrow xy=6\quad(\text{from }[1])\)

\(\displaystyle x+\frac6x=5\Rightarrow x^2-5x+6=0\Rightarrow x\in\{2,3\}\)

\(\displaystyle \text{As equations }[1]\text{ and }[2]\text{ are symmetrical, we have }(x,y)=(2,3),(3,2)\text{ as solutions.}\)