Question

Solve the following equations by Cramer's rule:
$3x+y+z=2,2x-4y+3z=-1,4x+y-3z=-11$

Answer 1

To solve the system by Cramer's rule, we do the following steps:

1. Write the system as a matrix equation $AX=b$ where:

$A=\left[\begin{array}{ccc}3& 1& 1\\ 2& -4& 3\\ 4& 1& -3\end{array}\right]$

$X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$b=\left[\begin{array}{c}2\\ -1\\ -11\end{array}\right]$

2. Next, we find the following determinants:

$\Delta =\left|\begin{array}{ccc}3& 1& 1\\ 2& -4& 3\\ 4& 1& -3\end{array}\right|=|A|=63$

${\Delta }_1=\left|\begin{array}{ccc}2& 1& 1\\ -1& -4& 3\\ -11& 1& -3\end{array}\right|=-63$

${\Delta }_2=\left|\begin{array}{ccc}3& 2& 1\\ 2& -1& 3\\ 4& -11& -3\end{array}\right|=126$

${\Delta }_3=\left|\begin{array}{ccc}3& 1& 2\\ 2& -4& -1\\ 4& 1& -11\end{array}\right|=189$

3. The solution is now given by:

$x=\frac{{\Delta }_1}{\Delta }=-1$

$y=\frac{{\Delta }_2}{\Delta }=2$

$z=\frac{{\Delta }_3}{\Delta }=3$