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\begin{document}

\title{Operations Research}
\author{Homework 2}
\date{Due in class Tuesday, September 19, 2017}
\maketitle

\begin{enumerate}

\item
Find an equation for the plane that contains the point
$\boldsymbol p=(2,4,6)$ and the line
\[
  \boldsymbol x =
  \begin{pmatrix}
  7 \\ 3 \\ 5
  \end{pmatrix}
  + \lambda \,
  \begin{pmatrix}
  -3 \\ 4 \\ 2
  \end{pmatrix} \,.
\]

\item  Find all solutions for the underdetermined linear system
$A\boldsymbol{x}=\boldsymbol{b}$, 
where
\begin{equation*}
  A =
  \begin{pmatrix}
    2 & 2 & 1 & 0 \\
    1 & 1 & 1 & 1 \\
    1 & 1 & 0 & -1 \\
    3 & 3 & 2 & 1
  \end{pmatrix}
  \qquad \text{and} \qquad
  \boldsymbol{b} =
  \begin{pmatrix}
    -1 \\ 1 \\ -2 \\ 0
  \end{pmatrix} \,.
\end{equation*}

\item Reconsider Problem~1 from Homework Set~1: \emph{Minimize}
\[
  z = 8 \, x + 12 \, y
\]
\emph{subject to}
\begin{gather*}
  5 \, x + 2 \, y \geq 20 \,, \\
  4 \, x + 3 \, y \geq 24 \,, \\
  y \geq 2 \,, \\
  x, y \geq 0 \,.
\end{gather*}

Introduce slack variables to write this linear programming
problem in the standard form: \emph{Minimize}
\begin{gather*}
  z = \c^T \x \\
\intertext{\emph{subject to}} 
  A\x = \b \, \\ 
  \x \geq 0
\end{gather*}
where the coefficients $\b, \c$, and the decision variables $\x$ are
written as column vectors, and $A$ is a matrix of matching dimension.




\end{enumerate}

\end{document}
