Problem 13: Determine whether the following system of equations are consistent or not. If yes, determine all solutions

 Problem 13: Determine whether the following system of equations are consistent or not. If yes, determine all solutions:

\begin{align*} x_1+x_2+2x_3=3\\ -x_1-3x_2+4x_3=2\\ -x_1-5x_2+10x_3=7 \end{align*}

Solution:

Consider the augmented matrix of this system and apply row operations.

\[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ -1 & -3& 4 & 2 \\ -1 & -5 & 10 & 7 \\ \end{array}\right] \xrightarrow{R2\rightarrow R_2+R_1,R3\rightarrow R_3+R_1} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ 0 & -2& 6 & 5 \\ 0 & -4 & 12 & 10 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-2R_2} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ 0 & -2& 6 & 5 \\ 0 & 0 & 0 & 0 \\ \end{array}\right] \xrightarrow{R_3\rightarrow -\frac{1}{2}R_3} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ 0 & 1& -3 & -\frac{5}{2} \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\] \[\xrightarrow{R1\rightarrow R_1-R_2} \left[\begin{array}{rrr|r} 1 & 0 & 5& \frac{11}{2} \\ 0 & 1& -3 & -\frac{5}{2} \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\]

Thus, rank of coefficient matrix = rank of augmented matrix = 2 $<$ number of unknowns. The system is consistent and it has infinitely many solutions given by

\begin{align*} {x}_{1}=\frac{1}{2}\left(11-10k\right),{x}_{2}=\frac{1}{2}\left(6k-5\right),{x}_{3}=k,k\in \mathbb{R} \end{align*}