Problem 12: Determine whether the following system of equations are consistent or not. If yes, determine all solutions

 Problem 12: Determine whether the following system of equations are consistent or not. If yes, determine all solutions:

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

Solution:

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

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

Thus, rank of the coefficient matrix = rank of augmented matrix = 3 - number of unknowns. This implies that the system is consistent and the solution is unique given by

\begin{align*} x_1=\frac{5}{3}\\ x_2=\frac{2}{3}\\ x_3=1 \end{align*}