Question

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

$x-4y=1$

$2x+3y=13$

Short answer

The required solution is $\left(5,1\right)$.

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 1 in place of a, 2 in place of c, $-4$ in place of b, 3 in place of d, 1 in place of e and 13 in place of f into role="math" localid="1647359635895" $x=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{cc}1& -4\\ 13& 3\end{array}\right|}{\left|\begin{array}{cc}1& -4\\ 2& 3\end{array}\right|}\\ =\frac{1\cdot 3-\left(-4\right)\cdot 13}{1\cdot 3-\left(-4\right)\cdot 2}\\ =\frac{55}{11}\\ =5\end{array}$

– 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 1 in place of a, 2 in place of c, $-4$ in place of b, 3 in place of d, 1 in place of e and 13 in place of f into role="math" localid="1647359841972" $y=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{cc}1& 1\\ 2& 13\end{array}\right|}{\left|\begin{array}{cc}1& -4\\ 2& 3\end{array}\right|}\\ =\frac{1\cdot 13-1\cdot 2}{1\cdot 3-\left(-4\right)\cdot 2}\\ =\frac{11}{11}\\ =1\end{array}$

Step 3- Write the solution

The obtained value of x is 5 and y is 1 so the solution is $\left(5,1\right)$.