Chapter 2.3, Problem 2P

Explanation of Solution

Indicating the solutions using Gauss-Jordan method:

From the given information, observe the following system of linear equations.

$\begin{array}{c}{x}_1+x_2+x_3=4\\ x_1+2x_2=6\end{array}$

The matrix form of the above system of linear equations is as follows:

$\left[A|b\right]=\left(\begin{array}{cccc}1& 1& 1& 4\\ 1& 2& 0& 6\end{array}\right)$

Start by replacing row 1 of $A|b-1\left(row\text{ 1 of }A|b\right)$+ row 2 of $A|b$. The result is as follows:

$A_1|b_1=\left(\begin{array}{cccc}0& -1& 1& -2\\ 1& 2& 0& 6\end{array}\right)$

Now, replace row 2 of $A_1|b_1\text{ - }1\left(row\text{ 2 of }{A}_1|b_1\right)$+ row 1 of $A_1|b_1$. The result is as follows:

$A_2|b_2=\left(\begin{array}{cccc}0& -1& 1& -2\\ 1& 1& 1& 4\end{array}\right)$

Now, replace row 2 of $A_2|b_2-1\left(row\text{ 2 of }{A}_2|b_2\right)$+ row 1 of $A_2|b_2$. The result is as follows:

$A_3|b_3=\left(\begin{array}{cccc}0& -1& 1& -2\\ 1& 0& 2& 2\end{array}\right)$

Then, replace row 2 of $A_3|b_3-1\left(row\text{ 2 of }{A}_3|b_3\right)$+ row 1 of $A_3|b_3$