Consider the system of equations
\[ \begin{array}{r l} x _ {1} - & x _ {2} + 2 x _ {3} = 1 \\ 2 x _ {1} & + 2 x _ {3} = 1 \\ x _ {1} - 3 x _ {2} + 4 x _ {3} = 2. \end{array} \]
Does this system have a solution? If so, describe explicitly all solutions.
Solution
The system is consistent and has the one-parameter family of solutions \[ (x_1,x_2,x_3) =\left(\frac12-t,-\frac12+t,t\right), \qquad t\in F. \]
1. Row reduction of the augmented matrix gives\[ \left[ \begin{array}{ccc|c} 1&-1&2&1\\ 2&0&2&1\\ 1&-3&4&2 \end{array} \right] \sim \left[ \begin{array}{ccc|c} 1&0&1&1/2\\ 0&1&-1&-1/2\\ 0&0&0&0 \end{array} \right]. \]
This is ordinary Gaussian elimination; in particular no row of the form \([0\ 0\ 0\mid c]\) with \(c\ne0\) occurs, so the system is consistent.
2. The displayed family is exactly the solution set.
The reduced equations are \[ x_1+x_3=\frac12, \qquad x_2-x_3=-\frac12. \] Set the free variable \(x_3=t\) and solve for \(x_1,x_2\).