Answer
$(\displaystyle \frac{3}{2},1)$
Work Step by Step
Cramer's rule
$\left\{\begin{array}{l}{a x+b y=s}\\{cx+dy=t}\end{array}\right.$
$D=\left|\begin{array}{ll}{a}&{b}\\{c}&{d}\end{array}\right|,D_{x}=\left|\begin{array}{ll}{s}&{b}\\{t}&{d}\end{array}\right|,D_{y}=\left|\begin{array}{ll}{a}&{s}\\{c}&{t}\end{array}\right|,$
If $D\displaystyle \neq 0,\qquad x=\frac{D_{x}}{D}\quad y=\frac{D_{y}}{D}$
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$\displaystyle \left\{\begin{array}{r}{2 x+3 y=6}\\{x-y=\frac{1}{2}}\end{array}\right.\Rightarrow\left[\begin{array}{ll}
a & b\\
c & d
\end{array}\right]=\displaystyle \left[\begin{array}{ll}
2 & 3\\
1 & -1
\end{array}\right],\quad \left[\begin{array}{l}
s\\
t
\end{array}\right]=\left[\begin{array}{l}
6\\
1/2
\end{array}\right]$
$\begin{array}{lllll}
D=\left|\begin{array}{ll}
2 & 3\\
1 & -1
\end{array}\right|= & & D_{x}=\left|\begin{array}{ll}
6 & 3\\
1/2 & -1
\end{array}\right|= & & D_{y}=\left|\begin{array}{ll}
2 & 6\\
1 & 1/2
\end{array}\right|=\\
=-2-3 & & =-6-3/2 & & =1-6\\
=-5\neq 0 & & =-15/2 & & =-5\\
& & & &
\end{array}$
$x=\displaystyle \frac{D_{x}}{D}=\frac{8}{40}=\frac{-15/2}{-5}=\frac{3}{2}$
$y=\displaystyle \frac{D_{y}}{D}=\frac{12}{40}=\frac{-5}{-5}=1$
Solution: $(x,y)=(\displaystyle \frac{3}{2},1)$
