Q. 38 - Precalculus Enhanced with Graphing Utilities

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.

$\left\{\begin{array}{l}x-y+2z=5\\ 3x+2y=4\\ -2x+2y-4z=-10\end{array}\right.$

Step-by-Step Solution

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 1. Given information

The given system of equations is   

$\left\{\begin{array}{l}x-y+2z=5\\ 3x+2y=4\\ -2x+2y-4z=-10\end{array}\right.$

Step 2. Determinant

The determinant D of the coefficients of the variables 

$$\begin{gathered} D=\left|\begin{array}{ccc}1& -1& 2\\ 3& 2& 0\\ -2& 2& -4\end{array}\right| \\ D={(-1)}^{1+1}\left(1\right)\left|\begin{array}{cc}2& 0\\ 2& -4\end{array}\right|+{(-1)}^{1+2}(-1)\left|\begin{array}{cc}3& 0\\ -2& -4\end{array}\right|+{(-1)}^{1+3}\left(2\right)\left|\begin{array}{cc}3& 2\\ -2& 2\end{array}\right| \\ D=1(-8-0)+1(-12-0)+2(6+4) \\ D=-8-12+20 \\ D=0 \end{gathered}$$

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

so Cramer's rule can be used to determine the solution of the system of equations.