Q. 53 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{l}-x+y+z=-1\\ -x+2y-3z=-4\\ 3x-2y-7z=0\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system is $x=5z-2, y=4z-3$where, z is any real number. 

Step 1. Given information.

The given system of equation are  

$\left\{\begin{array}{l}-x+y+z=-1\\ -x+2y-3z=-4\\ 3x-2y-7z=0\end{array}\right.$

Step 2. Calculation.

The augmented matrix of the system is: 

$\left[\begin{array}{cc}\begin{array}{ccc}-1& 1& 1\\ -1& 2& -3\\ 3& -2& -7\end{array}& \begin{array}{c}-1\\ -4\\ 0\end{array}\end{array}\right]$

Perform the row operations  $R_2=r_2-r_1; R_3=r_3+3r_1$:

$\left[\begin{array}{cc}\begin{array}{ccc}-1& 1& 1\\ 0& 1& -4\\ 0& 4& -16\end{array}& \begin{array}{c}-1\\ -3\\ -12\end{array}\end{array}\right]$

Perform the row operations  $R_3=r_3-4r_2$:

$\left[\begin{array}{cc}\begin{array}{ccc}-1& 1& 1\\ 0& 1& -4\\ 0& 0& 0\end{array}& \begin{array}{c}-1\\ -3\\ 0\end{array}\end{array}\right]$

The above matrix is in reduced row echelon form. The corresponding system of equations is 

$$\begin{gathered} -x+y+z=-1 \left(1\right) \\ y-4z=-3 \Rightarrow y=4z-3 \left(2\right) \end{gathered}$$

Step 3. Solve the equation.

Substitute $y=4z-3$ into equation (1).

$$\begin{gathered} -x+4z-3+z=-1 \\ -x+5z=2 \\ x=5z-2 \end{gathered}$$

Therefore, the solution of the system is $x=5z-2, y=4z-3$ where, z is any real number.

Step 4. Conclusion.

The solution of the system is $x=5z-2, y=4z-3$where, z is any real number.