System of equations Level 1

Exercise:

Solve in the following system

           $\left\{ \begin{align}
  & x+y=2 \\
 & {{2}^{-x}}+{{2}^{-y}}=1 \\
\end{align} \right.$

Solution: we have ${{2}^{-x}}+{{2}^{-y}}={{2}^{-x}}+{{2}^{-\left( 2-x \right)}}={{2}^{-x}}+{{2}^{-2+x}}=1$

So ${{2}^{-x}}+\frac{{{2}^{x}}}{2}=1\Leftrightarrow \frac{1}{{{2}^{x}}}+\frac{{{2}^{x}}}{{{2}^{2}}}=1\Leftrightarrow \frac{{{2}^{2}}+{{\left( {{2}^{x}} \right)}^{2}}}{{{2}^{x+2}}}=1\Leftrightarrow {{2}^{2}}+{{\left( {{2}^{x}} \right)}^{2}}={{2}^{x+2}}$

$\Rightarrow {{\left( {{2}^{x}} \right)}^{2}}-{{2}^{2}}\times {{2}^{x}}+{{2}^{2}}=0$

Put $t={{2}^{x}}\Leftrightarrow {{t}^{2}}-4t+4=0\Leftrightarrow {{\left( t-2 \right)}^{2}}=0\Leftrightarrow t=2$

If $t=2\Leftrightarrow {{2}^{x}}=2\Leftrightarrow x=1\,\,\And \,\,y=1$