Q. 35 - 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.

$\left\{\begin{array}{l}x+2y-z=-3\\ 2x-4y+z=-7\\ -2x+2y-3z=4\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of equations $\left\{\begin{array}{l}x+2y-z=-3\\ 2x-4y+z=-7\\ -2x+2y-3z=4\end{array}\right.$is  $\left(-3,\frac{1}{2},1\right)$

Step 1. Given information

The given system of equations is   

$\left\{\begin{array}{l}x+2y-z=-3\\ 2x-4y+z=-7\\ -2x+2y-3z=4\end{array}\right.$

Step 2. Determinant

The determinant D of the coefficients of the variables 

$$\begin{gathered} D=\left|\begin{array}{ccc}1& 2& -1\\ 2& -4& 1\\ -2& 2& -3\end{array}\right| \\ D={(-1)}^{1+1}\left(1\right)\left|\begin{array}{cc}-4& 1\\ 2& -3\end{array}\right|+{(-1)}^{1+2}\left(2\right)\left|\begin{array}{cc}2& 1\\ -2& -3\end{array}\right|+{(-1)}^{1+3}(-1)\left|\begin{array}{cc}2& -4\\ -2& 2\end{array}\right| \\ D=1(12-2)-2(-6+2)-1(4-8) \\ D=10+8+4 \\ D=22 \end{gathered}$$

$D\ne 0,$so Cramer's rule can be used to determine the solution of the system of equations.

so that $x=\frac{D_x}{D}, y=\frac{D_y}{D}, z=\frac{D_z}{D}$

Step 3. Value of D x

Determine $D_x$by replacing the coefficients of x in D with the constants 

$$\begin{gathered} D_x=\left|\begin{array}{ccc}-3& 2& -1\\ -7& -4& 1\\ 4& 2& -3\end{array}\right| \\ {D}_x={(-1)}^{1+1}(-3)\left|\begin{array}{cc}-4& 1\\ 2& -3\end{array}\right|+{(-1)}^{1+2}\left(2\right)\left|\begin{array}{cc}-7& 1\\ 4& -3\end{array}\right|+{(-1)}^{1+3}(-1)\left|\begin{array}{cc}-7& -4\\ 4& 2\end{array}\right| \\ {D}_x=-66 \end{gathered}$$

Step 4. Value of D y

Determine $D_y$by replacing the coefficients of y in D with the constants 

$$\begin{gathered} D_y=\left|\begin{array}{ccc}1& -3& -1\\ 2& -7& 1\\ -2& 4& -3\end{array}\right| \\ {D}_y={(-1)}^{1+1}\left(1\right)\left|\begin{array}{cc}-7& 1\\ 4& -3\end{array}\right|+{(-1)}^{1+2}(-3)\left|\begin{array}{cc}2& 1\\ -2& -3\end{array}\right|+{(-1)}^{1+3}(-1)\left|\begin{array}{cc}2& -7\\ -2& 4\end{array}\right| \\ {D}_y=11 \end{gathered}$$

Step 5. Value of D z

Determine $D_z$by replacing the coefficients of z in D with the constants 

$$\begin{gathered} D_z=\left|\begin{array}{ccc}1& 2& -3\\ 2& -4& -7\\ -2& 2& 4\end{array}\right| \\ {D}_z={(-1)}^{1+1}\left(1\right)\left|\begin{array}{cc}-4& -7\\ 2& 4\end{array}\right|+{(-1)}^{1+2}\left(2\right)\left|\begin{array}{cc}2& 1\\ -2& -3\end{array}\right|+{(-1)}^{1+3}(-1)\left|\begin{array}{cc}2& -7\\ -2& 4\end{array}\right| \\ {D}_z=22 \end{gathered}$$

Step 6. Solution of system

Solution for x  

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{-66}{22} \\ x=-3 \end{gathered}$$

Solution for y  

$$\begin{gathered} y=\frac{D_y}{D} \\ y=\frac{11}{22} \\ y=\frac{1}{2} \end{gathered}$$

Solution for z  

$$\begin{gathered} z=\frac{D_z}{D} \\ z=\frac{22}{22} \\ z=1 \end{gathered}$$

So the solution of the system of equations is  $\left(-3,\frac{1}{2},1\right)$