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{equation*} \begin{alignedat}{3} 2x \amp {}+{} \amp y \amp {}={} \amp 0 \\ x \amp {}+{} \amp 2y \amp {}={} \amp 3 \\ -2x \amp {}+{} \amp 2y \amp {}={} \amp 6 \\ \end{alignedat} \end{equation*}
  2. \begin{equation*} \begin{alignedat}{5} -x_1 \amp {}+{} \amp 2x_2 \amp \amp \amp {}+{} \amp x_3 \amp {}={} \amp 2 \\ 3x_1 \amp \amp \amp \amp \amp {}+{} \amp 2x_3 \amp {}={} \amp -1 \\ -x_1 \amp {}-{} \amp x_2 \amp \amp \amp {}+{} \amp x_3 \amp {}={} \amp 2 \\ \end{alignedat} \end{equation*}
  3. \begin{equation*} \begin{alignedat}{5} x_1 \amp {}+{} \amp 2x_2 \amp {}-{} \amp 5x_3 \amp {}-{} \amp x_4 \amp {}={} \amp -3 \\ -2x_1 \amp {}-{} \amp 2x_2 \amp {}+{} \amp 6x_3 \amp {}-{} \amp 2x_4 \amp {}={} \amp 4 \\ x_1 \amp \amp \amp {}-{} \amp x_3 \amp {}+{} \amp 9x_4 \amp {}={} \amp 7 \\ \amp \amp -x_2 \amp {}+{} \amp 2x_3 \amp {}-{} \amp x_4 \amp {}={} \amp 4 \\ \end{alignedat} \end{equation*}
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