Chapter 7.4, Problem 70E

To determine

To solve: the system of linear equation using Gauss Jordan elimination.

Answer to Problem 70E

The system of linear equation have an infinite number of solution of the form

  $\underset{\_}{x=1+2a}$ , $\underset{\_}{y=2+3a}$ and $\underset{\_}{z=a}$ where $a$ is any real number.

Explanation of Solution

Given information:

System of linear equation which need to be solved using Gauss Jordan elimination is as follows

  $\begin{array}{l}x+y-5z=3\\ x-2z=1\\ 2x-y-z=0\end{array}$ ..

Formula used:

According to Gauss Jordan elimination represent system of linear equation in augmented matrix and using row transformations, convert the obtained matrix into identity matrix to get the solution.

Calculation:

System of linear equation is as follows

  $\begin{array}{l}x+y-5z=3\\ x-2z=1\\ 2x-y-z=0\end{array}$ ..

Represent the system in augmented matrix

  $\left[\begin{array}{ccccc}1& 1& -5& :& 3\\ 1& 0& -2& :& 1\\ 2& -1& -1& :& 0\end{array}\right]$

Apply $R_2\to R_2-R_1$ and $R_3\to R_3-2R_1$

  $\left[\begin{array}{ccccc}1& 1& -5& :& 3\\ 0& -1& 3& :& -2\\ 0& -3& 9& :& -6\end{array}\right]$

Apply $R_2\to -R_2$ and $R_3\to \frac{R_3}{-3}$

  $\left[\begin{array}{ccccc}1& 1& -5& :& 3\\ 0& 1& -3& :& 2\\ 0& 1& -3& :& 2\end{array}\right]$

Apply $R_1\to R_1-R_2$ and $R_3\to R_3-R_2$

  $\left[\begin{array}{ccccc}1& 0& -2& :& 1\\ 0& 1& -3& :& 2\\ 0& 0& 0& :& 0\end{array}\right]$

Now represents the matrix in system of linear equation

  $\begin{array}{l}1\times x+0\times y-2\times z=1\\ 0\times x+1\times y-3\times z=2\\ 0\times x+0\times y+0\times z=0\end{array}$

  $\begin{array}{l}x-2z=1\\ y-3z=2\end{array}$

Solving $x$ and $y$ in term of $z$ ,

  $\begin{array}{l}x=1+2z\\ y=2+3z\end{array}$

Let $z=a$ where $a$ is any real number.

  $x=1+2a$

  $y=2+3a$

Solution set has the form $\left(1+2a,2+3a,a\right)$ which represents an infinite number of solutions.

Hence the system of linear equation have an infinite number of solution of the form $\underset{\_}{x=1+2a}$ , $\underset{\_}{y=2+3a}$ and $\underset{\_}{z=a}$ where $a$ is any real number.