Chapter 9, Problem 1MCCP

In Exercises 1-5, use matrices to find the complete solution to each system of equations, or show that none exists.

$\{\begin{array}{c}x+2y-3z=-7\\ 3x-y+2z=-8\\ 2x-y+z=5\end{array}$

To determine

To calculate: Augmented matrix for linear system $x+2y-3z=-7,3x-y+2z=8\text{ and }2x-y+z=5.$

Answer to Problem 1MCCP

Solution:

The augmented matrix is $\underset{\_}{\left[\begin{array}{llll}1\hfill & 2\hfill & -3\hfill & -7\hfill \\ 3\hfill & -1\hfill & 2\hfill & 8\hfill \\ 2\hfill & -1\hfill & 1\hfill & 5\hfill \end{array}\right]}$.

Explanation of Solution

Given:

The system of linear equation is:

$\begin{array}{ccc}x+2y-3z& =& -7\hfill \\ 3x-y+2z& =& 8\hfill \\ 2x-y+z& =& 5\hfill \end{array}$

Calculation:

The augmented matrix for a system of linear equations is:

$\begin{array}{ccc}{a}_{1}x+b_{1}y+c_{1}z& =& d_1\hfill \\ a_{2}x+b_{2}y+c_{2}z& =& d_2\hfill \\ a_{3}x+b_{3}y+c_{3}z& =& d_3\hfill \end{array}$

So, the augmented matrix is:

$\left[\begin{array}{llll}{a}_1\hfill & a_2\hfill & a_3\hfill & d_1\hfill \\ b_1\hfill & b_2\hfill & b_3\hfill & d_2\hfill \\ c_1\hfill & c_2\hfill & c_3\hfill & d_3\hfill \end{array}\right]$

The augmented matrix for given linear system can be written as given below,

$\left[\begin{array}{llll}1\hfill & 2\hfill & -3\hfill & -7\hfill \\ 3\hfill & -1\hfill & 2\hfill & 8\hfill \\ 2\hfill & -1\hfill & 1\hfill & 5\hfill \end{array}\right]$