Activity1.2.2Gaussian Elimination

Use Gaussian elimination to describe the solutions to the following systems of linear equations.

  1. Does the following linear system have exactly one solution, infinitely many solutions, or no solutions?

    \begin{equation*} \begin{alignedat}{4} x \amp {}+{} \amp y \amp {}+{} \amp 2z \amp {}={} \amp 1 \\ 2x \amp {}-{} \amp y \amp {}-{} \amp 2z \amp {}={} \amp 2 \\ -x \amp {}+{} \amp y \amp {}+{} \amp z \amp {}={} \amp 0 \\ \end{alignedat} \end{equation*}
  2. Does the following linear system have exactly one solution, infinitely many solutions, or no solutions?

    \begin{equation*} \begin{alignedat}{4} -x \amp {}-{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp -1 \\ 2x \amp {}+{} \amp 4y \amp {}-{} \amp z \amp {}={} \amp 5 \\ x \amp {}+{} \amp 2y \amp \amp \amp {}={} \amp 3 \\ \end{alignedat} \end{equation*}
  3. Does the following linear system have exactly one solution, infinitely many solutions, or no solutions?

    \begin{equation*} \begin{alignedat}{4} -x \amp {}-{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp -1 \\ 2x \amp {}+{} \amp 4y \amp {}-{} \amp z \amp {}={} \amp 5 \\ x \amp {}+{} \amp 2y \amp \amp \amp {}={} \amp 2 \\ \end{alignedat} \end{equation*}
in-context