\documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz,kgm}
\usepackage[top=0.56in]{geometry}

\newcommand{\kgmcomment}[1]{}

\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

\begin{document}

%\makeatletter
%\def\@oddfoot{\hfill \tikz \node[draw] {over} ; }
%\makeatother

\makeatletter
\def\@oddfoot{}
\makeatother

\newcommand{\dd}{\partial}

%\noindent
%\hfill Name: \underline{\hspace{3.5in}} 

%\begin{flushright}
%Name: \underline{\hspace{3.5in}} \\
%\end{flushright}
%\hrule

\newcommand{\vect}[1]{\mathbf{#1}}
\newcommand{\mb}[1]{\mathbf{#1}}
\newcommand{\Span}{\mathrm{Span}}

% Matrix Three By One
\newcommand{\MTBO}[3]%
{%
	\left[%
	\begin{array}{*{3}{r}}
	#1\\
	#2\\
	#3
	\end{array}%
	\right]%
}

% Matrix Three By One
\newcommand{\MTWBO}[2]%
{%
	\left[%
	\begin{array}{*{3}{r}}
	#1\\
	#2
	\end{array}%
	\right]%
}

%
% e.g. \kgmarray{num cols}{data}
%
\newcommand{\kgmarray}[2]%
{% 
    \begin{array}{*{#1}{r}}
        #2
    \end{array}
}

%
% e.g. \kgmmatrix{num cols}{data}
%
\newcommand{\kgmmatrix}[2]%
{% 
    \left[ \kgmarray{#1}{#2} \right]
}

%
% e.g. A \rrrule{$R2 \stackrel{+}{=} (-3)R1$ \\ $R3 \stackrel{+}{=} R1$}
%
\newcommand{\rrrule}[1]%
{%
    \begin{tikzpicture}
        \scriptsize
        \node[rectangle,align=left] (TMP) {#1} ;
        \path (TMP.south west) ++(-0.3ex, -0.3ex) node[coordinate] (TMPLEFT) {} ;
        \path (TMP.south east) ++(0.3ex, -0.3ex) node[coordinate] (TMPRIGHT) {} ;
        \draw[->] (TMPLEFT) -- (TMPRIGHT) ;
    \end{tikzpicture}
}      

\newcommand{\RR}{\mathbb{R}}

\setlength{\columnseprule}{0.4pt}

\pagestyle{empty}

~
\vspace{2cm}

\noindent
\textbf{Directions:}  
\begin{enumerate}
	\item Write your name with one character in each box below.
	\item Show all work.  No credit for answers without work.
\end{enumerate}

\newpage
\headers{Math 343}{Test 1}{Sep. 19, 2025}

~
\vspace{1.5cm}

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item\points{15} Solve the following system of equations.
\[\kgmarray{7}{%
	  x_1 &-&   x_2 &-&    x_3   &=&   0\\
   -30x_1 &+& 36x_2 &-&   7x_3   &=&   7\\
	-5x_1 &+&  6x_2 &-&    x_3   &=&   1
}\]
\vfill

\item\points{15} Give an equation for the components of $\vect{b}$ that determines when the system $A\vect{x}=\vect{b}$ is consistent.
\begin{align*}
A &= \kgmmatrix{3}
{ 
	0	& 1	& 2 \\
	1	& 1	& 7 \\
	-2	& -2	& -14
} & 
\vect{b}=\MTBO{b_1}{b_2}{b_3}
\end{align*}
\vfill


\newpage
~
\vspace{1.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item\points{15} List the elementary row operations and give a brief description of each.
\vspace{4cm}

\item\points{15} Determine which values of $h$, if any, make the linear system represented by the following augmented matrix have infinitely many solutions.
\[ \kgmmatrix{4}
{
    3 &  -2  &  5 &   1\\
    2 &   h  &  2 &  -1\\
    1 &  -1  &  3 &   2
}
\]


\newpage
~
\vspace{1.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item\points{15} Given the augmented matrix, express the solution set in parametric form.
\begin{align*}
\kgmmatrix{8}{
1 & 5 & 0 & 0 & -7 &  4 & 0 & 1\\
0 & 0 & 1 & 0 &  0 &  2 & 0 & -1\\
0 & 0 & 0 & 1 &  6 & -1 & 0 & 0\\
0 & 0 & 0 & 0 &  0 &  0 & 1 & -1
}
\end{align*}
\vspace{4cm}

\item\ppoints{5}{2} True/False.  Justify your answers.
\begin{enumerate}
    \item Every matrix is row-equivalent to infinitely many matrices.
	\item Every inconsistent linear system can be made consistent by deleting a carefully chosen equation.
	\item If $A\mb{x}=\mb{b}$ and $A\mb{x}=\mb{c}$ are consistent systems, then the solution sets are translations of one another.
	\item If $\mb{a}_1,\ldots,\mb{a}_n\in \RR^m$ and $n\ge m$, then $\Span(\{\mb{a}_1,\ldots,\mb{a}_n\}) = \RR^m$.
	\item If $\mb{u}$ is a scalar multiple of $\mb{a}$ and $\mb{v}$ is a scalar multiple of $\mb{b}$, then $\mb{u}+\mb{v}$ is a scalar multiple of $\mb{a}+\mb{b}$.
\end{enumerate}
\vfill



\newpage
~
\vspace{1.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item An economy has $3$ sectors: tech, food, and energy.  The output of the tech sector is consumed as follows: $4/5$ to tech, $1/10$ to food, and $1/10$ to energy.  The output of the food sector is consumed as follows: $1/3$ to tech, $1/6$ to food, and $1/2$ to energy.  The output of the energy sector is consumed as follows: $1/4$ to tech, $1/4$ to food, and $1/2$ to energy.    
\begin{enumerate}
    \item\points{10} Let $p_t$, $p_f$, and $p_e$, be the total cost (equivalently, the total value) in billions of the food, energy, and tech sectors, respectively.  Assuming  that the total cost (or total value) of each sector equals the that sector's  total expenses, obtain a linear system in variables $p_t$, $p_f$, and $p_e$.
    \vspace{4cm}
    \item\points{5} Given that the total value of the economy is \$120 billion (i.e. $p_t+p_f+p_e = 120$), find the value of each sector.  (Hint: first scale each row to eliminate fractions and then try to avoid reintroducing them.)
    \vfill
\end{enumerate}
\end{enumerate}
\newpage
~
\vspace{1.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page Scratch
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~

\end{document}
