Question

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x-2y+3z=1\\ 3x+y-2z=0\\ 2x-4y+6z=2\end{array}\right.$

Short answer

Determinant of the coefficients of the variables of the system$D=0,$so Cramer's rule can not be used to determine the solution of the system of equations.

Step-by-step solution

Step 1. Given information   

The given system of equations is

$\left\{\begin{array}{l}x-2y+3z=1\\ 3x+y-2z=0\\ 2x-4y+6z=2\end{array}\right.$

Step 2. Determinant   

The determinant D of the coefficients of the variables

$$\begin{gathered} D=\left|\begin{array}{ccc}1& -2& 3\\ 3& 1& -2\\ 2& -4& 6\end{array}\right| \\ D={(-1)}^{1+1}\left(1\right)\left|\begin{array}{cc}1& -2\\ -4& 6\end{array}\right|+{(-1)}^{1+2}(-2)\left|\begin{array}{cc}3& -2\\ 2& 6\end{array}\right|+{(-1)}^{1+3}\left(3\right)\left|\begin{array}{cc}3& 1\\ 2& -4\end{array}\right| \\ D=1(6-8)+2(18+4)+3(-12-2) \\ D=-2+44-42 \\ D=0 \end{gathered}$$

$D=0,$so the system of equations is either inconsistent or have infinitely many solutions

so Cramer's rule can not be used to determine the solution of the system of equations.