# Solve the following equations

Question:

If $e\left[\begin{array}{ll}e^{x} & e^{y} \\ e^{y} & e^{x}\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$, then $x=$__________$y=$_________

Solution:

$e\left[\begin{array}{ll}e^{x} & e^{y} \\ e^{y} & e^{x}\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}e^{x} \times e & e^{y} \times e \\ e^{y} \times e & e^{x} \times e\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}e^{x+1} & e^{y+1} \\ e^{y+1} & e^{x+1}\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$

$\Rightarrow e^{x+1}=1$ and $e^{y+1}=1$

$\Rightarrow e^{x+1}=e^{0}$ and $e^{y+1}=e^{0}$

$\Rightarrow x+1=0$ and $y+1=0$

$\Rightarrow x=-1$ and $y=-1$

If $e\left[\begin{array}{ll}e^{x} & e^{y} \\ e^{y} & e^{x}\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$, then $x=$ ___$-1$______$y=$____$-1$