91Ó°ÊÓ

Apply the Cramer’s rule for the value of x.

$x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute 3 in place of a, $\frac{1}{2}$ in place of c, $-2$ in place of b, $-\frac{2}{3}$ in place of d, 4 in place of e and 1 in place of f into role="math" localid="1647240268068" $x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}4& −2\\ 1& −\frac{2}{3}\end{array}\right|}{\left|\begin{array}{cc}3& −2\\ \frac{1}{2}& −\frac{2}{3}\end{array}\right|}\\ =\frac{4â‹\dots \left(−\frac{2}{3}\right)−\left(−2\right)â‹\dots 1}{3â‹\dots \left(−\frac{2}{3}\right)−\left(−2\right)â‹\dots \frac{1}{2}}\\ =\frac{−\frac{2}{3}}{−1}\\ =\frac{2}{3}\end{array}$

Apply the Cramer’s rule for the value of y.

$y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute 3 in place of a, $\frac{1}{2}$ in place of c, $-2$ in place of b, $-\frac{2}{3}$ in place of d, 4 in place of e and 1 in place of f into role="math" localid="1647240451038" $y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}3& 4\\ \frac{1}{2}& 1\end{array}\right|}{\left|\begin{array}{cc}3& −2\\ \frac{1}{2}& −\frac{2}{3}\end{array}\right|}\\ =\frac{3â‹\dots 1−4â‹\dots \left(\frac{1}{2}\right)}{3â‹\dots \left(−\frac{2}{3}\right)−\left(−2\right)â‹\dots \frac{1}{2}}\\ =\frac{1}{−1}\\ =−1\end{array}$

The obtained value of x is $\frac{2}{3}$ and y is $-1$ so the solution is $\left(\frac{2}{3},-1\right)$.