Question

Solve the following equation using Cramer's Rule:
$x-2y=3,-x+3z=5,4y-z=1$

Answer 1

Given $x-2y=3-x+3z=54y-z=1$

coefficient matrix

$D=\left[\begin{array}{ccc}1& -2& 0\\ -1& 0& 3\\ 0& 4& -1\end{array}\right]$

x matrix

$D_x=\left[\begin{array}{ccc}3& -2& 0\\ 5& 0& 3\\ 1& 4& -1\end{array}\right]$

y matrix

$D_y=\left[\begin{array}{ccc}1& 3& 0\\ -1& 5& 3\\ 0& 1& -1\end{array}\right]$

z matrix

$D_z=\left[\begin{array}{ccc}1& -2& 3\\ -1& 0& 5\\ 0& 4& 1\end{array}\right]$

$\left|D\right|=-10\left|D_x\right|=-52\left|D_y\right|=-11\left|D_z\right|=-34x=\frac{-52}{-10}=\frac{26}{5}y=\frac{-11}{-10}=\frac{11}{10}z=\frac{-34}{-10}=\frac{17}{5}$