Consider the system of equations

\[\begin{matrix} x_{1} & & & + & 3 \, x_{3} & = & -4 \\ & & x_{2} & + & 3 \, x_{3} & = & -5 \\ x_{1} & & & + & 4 \, x_{3} & = & -6 \\ 3 \, x_{1} & - & 2 \, x_{2} & + & 6 \, x_{3} & = & -8 \\ \end{matrix}\]

  1. Write an augmented matrix corresponding to this system.
  2. Write a vector equation corresponding to this system.

Answer:

  1. \[ \left[\begin{array}{ccc|c} 1 & 0 & 3 & -4 \\ 0 & 1 & 3 & -5 \\ 1 & 0 & 4 & -6 \\ 3 & -2 & 6 & -8 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 3 \\ 4 \\ 6 \end{array}\right] = \left[\begin{array}{c} -4 \\ -5 \\ -6 \\ -8 \end{array}\right] \]