Question

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

$1.5m-0.7n=0.5$

$2.2m-0.6n=-7.4$

Short answer

The required solution is $\left(-8.5625,-19.0625\right)$.

Step-by-step solution

Step 1- Determine the value of m

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

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

Substitute $1.5$ in place of a, $2.2$ in place of c, $-0.7$ in place of b, $-0.6$ in place of d, $0.5$ in place of e and role="math" localid="1647193225857" $-7.4$ in place of f into role="math" localid="1647193212588" $m=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of m.

$\begin{array}{c}m=\frac{\left|\begin{array}{cc}0.5& -0.7\\ -7.4& -0.6\end{array}\right|}{\left|\begin{array}{cc}1.5& -0.7\\ 2.2& -0.6\end{array}\right|}\\ =\frac{0.5\cdot \left(-0.6\right)-\left(-7.4\right)\cdot \left(-0.7\right)}{1\cdot 5\cdot \left(-0.6\right)-2.2\cdot \left(-0.7\right)}\\ =\frac{-5.48}{0.64}\\ =-8.5625\end{array}$

– Determine the value of n 

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

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

Substitute $1.5$ in place of a, $2.2$ in place of c, $-0.7$ in place of b, $-0.6$ in place of d, role="math" localid="1647193438046" $0.5$ in place of e and role="math" localid="1647193431538" $-7.4$ in place of f into role="math" localid="1647193422851" $n=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of n.

$\begin{array}{c}m=\frac{\left|\begin{array}{cc}1.5& 0.5\\ 2.2& -7.4\end{array}\right|}{\left|\begin{array}{cc}1.5& -0.7\\ 2.2& -0.6\end{array}\right|}\\ =\frac{1.5\cdot \left(-7.4\right)-2.2\cdot 0.5}{1\cdot 5\cdot \left(-0.6\right)-2.2\cdot \left(-0.7\right)}\\ =\frac{-12.2}{0.64}\\ =-19.0625\end{array}$

Step 3- Write the solution

The obtained value of m is $-8.5625$ and n is $-19.0625$ so the solution is $\left(-8.5625,-19.0625\right)$.