Activity 1.2.3 Augmented matrices and solution spaces

Write the augmented matrix for the system of equations
\begin{equation*} \begin{alignedat}{4} x \amp {}+{} \amp 2y \amp {}{} \amp z \amp {}={} \amp 1 \\ 3x \amp {}+{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp 7 \\ x \amp \amp \amp {}+{} \amp 4z \amp {}={} \amp 3 \\ \end{alignedat} \end{equation*}and perform Gaussian elimination to describe the solution space of the system of equations in as much detail as you can.

Suppose that you have a system of linear equations in the unknowns \(x\) and \(y\) whose augmented matrix is row equivalent to
\begin{equation*} \left[ \begin{array}{rrr} 1 \amp 0 \amp 3 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \\ \end{array} \right]. \end{equation*}Write the system of linear equations corresponding to the augmented matrix. Then describe the solution set of the system of equations in as much detail as you can.

Suppose that you have a system of linear equations in the unknowns \(x\) and \(y\) whose augmented matrix is row equivalent to
\begin{equation*} \left[ \begin{array}{rrr} 1 \amp 0 \amp 3 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array} \right]. \end{equation*}Write the system of linear equations corresponding to the augmented matrix. Then describe the solution set of the system of equations in as much detail as you can.

Suppose that the augmented matrix of a system of linear equations has the following shape where \(*\) could be any real number.
\begin{equation*} \left[ \begin{array}{rrrrrr} * \amp * \amp * \amp * \amp * \amp * \\ * \amp * \amp * \amp * \amp * \amp * \\ * \amp * \amp * \amp * \amp * \amp * \\ \end{array} \right]. \end{equation*}How many equations are there in this system and how many unknowns?
Based on our earlier discussion in SectionÂ 1.1, do you think it's possible that this system has exactly one solution, infinitely many solutions, or no solutions?

Suppose that this augmented matrix is row equivalent to
\begin{equation*} \left[ \begin{array}{rrrrrr} 1 \amp 2 \amp 0 \amp 0 \amp 3 \amp 2 \\ 0 \amp 0 \amp 1 \amp 2 \amp 1 \amp 1 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right]. \end{equation*}Make a choice for the names of the unknowns and write the corresponding system of linear equations. Does the system have exactly one solution, infinitely many solutions, or no solutions?