Q. 24 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{r}x-y+z=0\\ x-y-5z-6=0\\ 2x-2y+z-1=0\end{array}\right.$

Step-by-Step Solution

Answer

The solution is $\left\{\left(x,y,z\right)|x=y+1,z=-1, y is any real number\right\}$.

Step 1. Given information

The given system of equations is $\left\{\begin{array}{r}x-y+z=0\\ x-y-5z-6=0\\ 2x-2y+z-1=0\end{array}\right.$

Step 2. Solve system of equations using matrix

Write the augmented matrix that represents the system of equations

$\left[\begin{array}{rrrr}1& -1& 1& 0\\ 1& -1& -5& 6\\ 2& -2& 1& 1\end{array}\right]$

Perform the operations $R_2=R_2-R_1$,

$\left[\begin{array}{cccc}1& -1& 1& 0\\ 1-1& -1-(-1)& -5-\left(1\right)& 6-0\\ 2& -2& 1& 1\end{array}\right]\to \left[\begin{array}{crrc}1& -1& 1& 0\\ 0& 0& -6& 6\\ 2& -2& 1& 1\end{array}\right]$

Now, perform the operations $R_3=R_3-2R_1$,

$\left[\begin{array}{cccc}1& -1& 1& 0\\ 0& 0& -6& 6\\ 2-2\left(1\right)& -2-2(-1)& 1-2\left(1\right)& 1-2\left(0\right)\end{array}\right]\to \left[\begin{array}{rrrr}1& -1& 1& 0\\ 0& 0& -6& 6\\ 0& 0& -1& 1\end{array}\right]$

The system of equations corresponding to this matrix is

$$\begin{gathered} \left\{x-y+z=0 \\ -6z=6 \\ -z=1\right. \end{gathered}$$

So,

$$\begin{gathered} z=-1 \\ x-y+\left(-1\right)=0 \\ x-y=1 \\ x=y+1 \end{gathered}$$

Therefore, the solution is $\left\{\left(x,y,z\right)|x=y+1,z=-1, y is any real number\right\}$