Directions: You may work to solve these problems in groups, but all written work must be your own. Show all work; no credit for solutions without work..

1. Solve the following systems of linear equations. Verify that your solution is correct by substituting values back into their original equations.

   1.  
      \[\begin {array}{*{5}{r}} x_1 & - & x_2 & = & 10\\ 3x_1 & + & 2x_2 & = & -5 \end {array}\]
   2.  
      \[\begin {array}{*{7}{r}} x_1 & + & x_2 & - & 2x_3 & = & 0 \\ & & 3x_2 & + & 2x_3 & = & 1 \\ 2x_1 & & & - & x_3 & = & 8 \end {array}\]

2. [1.1.{19,20}] Determine if the system is consistent. Do not fully solve.

   1. \[ \begin {array}{*{9}{r}} x_1 && &+& 3x_3 && &=& 2\\ && x_2 && &-& 3x_4 &=& 3\\ && -2x_2 &+& 3x_3 &+& 2x_4 &=& 1\\ 3x_1 && && &+& 7x_4 &=& -5 \end {array} \]
   2. \[ \begin {array}{*{9}{r}} x_1 && && &-& 2x_4 &=& -3\\ && 2x_2 &+& 2x_3 && &=& 0\\ && && x_3 &+& 3x_4 &=& 1\\ -2x_1 &+& 3x_2 &+& 2x_3 &+& x_4 &=& 5 \end {array} \]

3. An augmented matrix with unknown entries.

   1. Find an equation involving \(a\), \(b\), and \(c\) that makes the following augmented matrix correspond to a consistent system. \[ \left [ \begin {array}{ccc} 1 & 2 & 4 \\ 2 & 1 & 5 \\ a & b & c \end {array} \right ] \]
   2. Let \(\mathcal {L}\) be the set of lines \(ax_1 + bx_2 = c\) such that \(a\), \(b\), and \(c\) make the augmented matrix in part (a) consistent. Give a geometric description of \(\mathcal {L}\).