Q13. - Algebra 2

Question

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

$2m+7n=4$

$m-2n=-20$

Step-by-Step Solution

Answer

The required solution is $\left(-\frac{148}{3},-\frac{44}{3}\right)$.

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 2 in place of a, 1 in place of c, 7 in place of b, $-2$ in place of d, 4 in place of e and $-20$ in place of f into $m=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of m.

$\begin{array}{c}m=\frac{\left|\begin{array}{cc}4& -7\\ -20& -2\end{array}\right|}{\left|\begin{array}{cc}2& -7\\ 1& -2\end{array}\right|}\\ =\frac{4\cdot \left(-2\right)-\left(-7\right)\cdot \left(-20\right)}{2\cdot \left(-2\right)-\left(-7\right)\cdot 1}\\ =\frac{-148}{3}\end{array}$

Step 2 – 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 2 in place of a, 1 in place of c, 7 in place of b, $-2$ in place of d, 4 in place of e and $-20$ in place of f into $n=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of n.

$\begin{array}{c}n=\frac{\left|\begin{array}{cc}2& 4\\ 1& -20\end{array}\right|}{\left|\begin{array}{cc}2& -7\\ 1& -2\end{array}\right|}\\ =\frac{2\cdot \left(-20\right)-\left(4\right)\cdot 1}{2\cdot \left(-2\right)-\left(-7\right)\cdot 1}\\ =\frac{-44}{3}\end{array}$

Step 3- Write the solution

The obtained value of m is $-\frac{148}{3}$ and n is $-\frac{44}{3}$ so the solution is $\left(-\frac{148}{3},-\frac{44}{3}\right)$.