Question

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

$2r-s=1$

$3r+2s=19$

Short answer

The required solution is $\left(3,5\right)$.

Step-by-step solution

Step 1- Determine the value of r

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

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

Substitute 2 in place of a, 3 in place of c,$-1$ in place of b, 2 in place of d, 1 in place of e and 19 in place of f into role="math" localid="1647189600339" $r=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of r.

$\begin{array}{c}r=\frac{\left|\begin{array}{cc}1& -1\\ 19& 2\end{array}\right|}{\left|\begin{array}{cc}2& -1\\ 3& 2\end{array}\right|}\\ =\frac{1\cdot 2-\left(-1\right)\cdot 19}{2\cdot 2-\left(-1\right)\cdot 3}\\ =\frac{21}{7}\\ =3\end{array}$

– Determine the value of s 

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

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

Substitute 2 in place of a, 3 in place of c, $-1$ in place of b, 2 in place of d, 1 in place of e and 19 in place of f into role="math" localid="1647189747064" $s=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of s.

$\begin{array}{c}s=\frac{\left|\begin{array}{cc}2& 1\\ 3& 19\end{array}\right|}{\left|\begin{array}{cc}2& -1\\ 3& 2\end{array}\right|}\\ =\frac{2\cdot 19-1\cdot 3}{2\cdot 2-\left(-1\right)\cdot 3}\\ =\frac{35}{7}\\ =5\end{array}$

Step 3- Write the solution

The obtained value of r is 3 and s is 5 so the solution is $\left(3,5\right)$.