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\headers{Math343}{Homework 1}{Due 27 Aug 2025}

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\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work..

\begin{enumerate}

\item Solve the following systems of linear equations.  Verify that your solution is correct by substituting values back into their original equations.
\begin{enumerate}
	\item ~\\
	\[\begin{array}{*{5}{r}}
	 x_1  & - & x_2  & = & 10\\
	 3x_1 & + & 2x_2 & = & -5
	 \end{array}\]
	\item ~\\
	\[\begin{array}{*{7}{r}}
	 x_1 & + & x_2  & - & 2x_3 & = & 0 \\
	     &   & 3x_2 & + & 2x_3 & = & 1 \\
	2x_1 &   &      & - &  x_3 & = & 8
	\end{array}\]
\end{enumerate}

\item {}[1.1.\{19,20\}] Determine if the system is consistent.  Do not fully solve.
\begin{enumerate}
\item
\[ \begin{array}{*{9}{r}}
	x_1  &&       &+& 3x_3 &&       &=& 2\\
	     && x_2   &&       &-& 3x_4 &=& 3\\ 
	     && -2x_2 &+& 3x_3 &+& 2x_4 &=& 1\\
	3x_1 &&       &&       &+& 7x_4 &=& -5
 \end{array}
 \]
\item
\[ \begin{array}{*{9}{r}}
	x_1   &&       &&       &-& 2x_4 &=& -3\\
	      && 2x_2  &+& 2x_3 &&       &=& 0\\ 
	      &&       &&   x_3 &+& 3x_4 &=& 1\\
	-2x_1 &+& 3x_2 &+& 2x_3 &+&  x_4 &=& 5
 \end{array}
 \]
\end{enumerate}

\item An augmented matrix with unknown entries.
\begin{enumerate}
\item Find an equation involving $a$, $b$, and $c$ that makes the following augmented matrix correspond to a consistent system.
\[
\left[
\begin{array}{ccc}
1 & 2 & 4 \\
2 & 1 & 5 \\
a & b & c
\end{array}
\right]
\]
\item Let $\mathcal{L}$ be the set of lines $ax_1 + bx_2 = c$ such that $a$, $b$, and $c$ make the augmented matrix in part (a) consistent.  Give a geometric description of $\mathcal{L}$.
\end{enumerate}


\end{enumerate}
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