Question

Use Cramer’s Rule to solve each system of equations.

$5x+2y=8$

$2x-3y=7$

Short answer

The required solution is $\left(2,-1\right)$.

Step-by-step solution

Step 1- Determine the value of x

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 5 in place of a, 2 in place of c, 2 in place of b, $-3$in place of

d, 8 in place of e and 7 in place of f intorole="math" localid="1647187883563" $x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

and perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}8& 2\\ 7& -3\end{array}\right|}{\left|\begin{array}{cc}5& 2\\ 2& -3\end{array}\right|}\\ =\frac{8\cdot \left(-3\right)-2\cdot 7}{5\cdot \left(-3\right)-2\cdot 2}\\ =\frac{-38}{-19}\\ =2\end{array}$

– Determine the value of y 

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 5 in place of a, 2 in place of c, 2 in place of b, $-3$in place of

d, 8 in place of e and 7 in place of f intorole="math" localid="1647188074242" $y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

and perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}5& 8\\ 2& 7\end{array}\right|}{\left|\begin{array}{cc}5& 2\\ 2& -3\end{array}\right|}\\ =\frac{5\cdot 7-8\cdot 2}{5\cdot \left(-3\right)-2\cdot 2}\\ =\frac{19}{-19}\\ =-1\end{array}$

Step 3- Write the solution

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