Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 1 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} 7 \\ 4 \\ 8 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -3 \\ -6 \end{array}\right] = \left[\begin{array}{c} 4 \\ 3 \\ 6 \end{array}\right] \]

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

Answer:

  1. \[\begin{matrix} x_{1} & - & 3 \, x_{2} & + & 7 \, x_{3} & - & 4 \, x_{4} & = & 4 \\ x_{1} & - & 2 \, x_{2} & + & 4 \, x_{3} & - & 3 \, x_{4} & = & 3 \\ 2 \, x_{1} & - & 4 \, x_{2} & + & 8 \, x_{3} & - & 6 \, x_{4} & = & 6 \\ \end{matrix}\]

  2. \[ \left[\begin{array}{cccc|c} 1 & -3 & 7 & -4 & 4 \\ 1 & -2 & 4 & -3 & 3 \\ 2 & -4 & 8 & -6 & 6 \end{array}\right] \]