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Section C.1 foo

Consider the scalar system of equations

\begin{alignat*}{5} 3x_1 &\,+\,& 2x_2 &\,\,& &\,+\,&x_4 &= 1 \\ -x_1 &\,-\,& 4x_2 &\,+\,&x_3&\,-\,&7x_4 &= 0 \\ &\,\,& x_2 &\,-\,&x_3 &\,\,& &= -2 \end{alignat*}

  1. Rewrite this system as a vector equation.

  2. Write an augmented matrix corresponding to this system.

Solution.

  1. \begin{equation*} x_1\left[\begin{array}{c} 3 \\ -1 \\ 0 \end{array}{}\right] + x_2 \left[\begin{array}{c}2 \\ -4 \\ 1 \end{array}{}\right]+ x_3 \left[\begin{array}{c} 1 \\ 1 \\ -1 \end{array}{}\right] + x_4 \left[\begin{array}{c} 1 \\ -7 \\ 0 \end{array}{}\right] = \left[\begin{array}{c} 1 \\ 0 \\ -2 \end{array}{}\right] \end{equation*}

  2. \begin{equation*} \left[\begin{array}{cccc|c} 3 & 2 & 0 & 1 & 1 \\ -1 & -4 & 1 & -7 & 0 \\ 0 & 1 & -1 & 0 & -2 \end{array}\right] \end{equation*}