Question

Using Cramer’s Rule, what is the value of y in the system of linear equations below?

Answer 1

for the system of equations

$a_1x+b_1y=c_1 \\a_2x+b_2y=c_2$

the three matrices needed to use Cramer's Rule are:

$D=\left[\begin{array}{cc}a_1&b_1\\a_2&b_2\end{array}\right] \\\\D_x=\left[\begin{array}{cc}c_1&b_1\\c_2&b_2\end{array}\right] \\\\D_y=\left[\begin{array}{cc}a_1&c_1\\a_2&c_2\end{array}\right]$.

To use Cramer's Rule we have to calculate the three determinants listed below of the matrices listed below.

$D=\left[\begin{array}{cc}2&5\\-3&-2\end{array}\right] \\\\D_x=\left[\begin{array}{cc}-13&5\\3&-2\end{array}\right] \\\\D_y=\left[\begin{array}{cc}2&-13\\-3&3\end{array}\right]$ .

The value of the determinants are shown below.

$det(D)=(2)(-2)-(-3)(5)=-4+15=11\\det(D_x)=(-13)(-2)-(3)(5)=26-15=11\\det(D_y)=(2)(3)-(-3)(-13)=6-39=-33$

The value of y is $\frac{det(D_y)}{det(D)}=-\frac{33}{11} =-3.$.