Answer
$x=5,\quad y=1$
Work Step by Step
Matrices are equal if they have the same dimension and
all the corresponding entries are equal.
$\left[\begin{array}{ll}
x & y\\
-y & x
\end{array}\right]-\left[\begin{array}{ll}
y & x\\
x & -y
\end{array}\right]=\left[\begin{array}{ll}
x-y & y-x\\
-y-x & x+y \\
&
\end{array}\right]$
$\left[\begin{array}{ll}
x-y & y-x\\
-y-x & x+y
\end{array}\right]$=$\left[\begin{array}{ll}
4 & -4\\
-6 & 6
\end{array}\right]$
$a_{11}\Rightarrow\quad x-y=4\Rightarrow\quad x=y+4$
Insert this into:
$a_{22}\Rightarrow\quad x+y=6\Rightarrow\quad y+4+y=6\Rightarrow\quad 2y=2\Rightarrow\quad y=1$
Back substituting, $x=y+4=1+4=5$
Checking at $a_{12},$
$1-5=-4$
Checking at $a_{21},$
$-5-1=-6$
$x=5,\quad y=1$
