Exercise 1

For each of the linear systems below, write the associated augmented matrix and find the reduced row echelon matrix that is row equivalent to it. Identify the basic and free variables and then describe the solution space of the original linear system using a parametric description, if appropriate.

1. $$\begin{alignedat}{3} 2x & {}+{} & y & {}={} & 0 \\ x & {}+{} & 2y & {}={} & 3 \\ -2x & {}+{} & 2y & {}={} & 6 \\ \\ \end{alignedat}$$

2. $$\begin{alignedat}{5} -x_1 & {}+{} & 2x_2 & & & {}+{} & x_3 & {}={} & 2 \\ 3x_1 & & & & & {}+{} & 2x_3 & {}={} & -1 \\ -x_1 & {}-{} & x_2 & & & {}+{} & x_3 & {}={} & 2 \\ \\ \end{alignedat}$$

3. $$\begin{alignedat}{5} x_1 & {}+{} & 2x_2 & {}-{} & 5x_3 & {}-{} & x_4 & {}={} & -3 \\ -2x_1 & {}-{} & 2x_2 & {}+{} & 6x_3 & {}-{} & 2x_4 & {}={} & 4 \\ x_1 & & & {}-{} & x_3 & {}+{} & 9x_4 & {}={} & -7 \\ & & -x_2 & {}+{} & 2x_3 & {}-{} & x_4 & {}={} & 4 \\ \\ \end{alignedat}$$