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<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
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<a href="E.html" data-scroll="E"><span class="codenumber">1</span> <span class="title">Systems of Linear Equations (E)</span></a><ul>
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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="E1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">1.1</span> <span class="title">Linear Systems, Vector Equations, and Augmented Matrices (E1)</span>
</h2>
<article class="definition definition-like" id="definition-1"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.1.1</span><span class="period">.</span>
</h6>
<p id="p-1">A <dfn class="terminology">linear equation</dfn> is an equation of the variables \(x_i\) of the form</p>
<div class="displaymath">
\begin{equation*}
a_1x_1+a_2x_2+\dots+a_nx_n=b\text{.}
\end{equation*}
</div>
<p id="p-2">A <dfn class="terminology">solution</dfn> for a linear equation is a Euclidean vector</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
s_1 \\
s_2 \\
\vdots \\
s_n
\end{bmatrix}
\end{equation*}
</div>
<p data-braille="continuation">that satisfies</p>
<div class="displaymath">
\begin{equation*}
a_1s_1+a_2s_2+\dots+a_ns_n=b
\end{equation*}
</div>
<p data-braille="continuation">(that is, a Euclidean vector that can be plugged into the equation).</p></article><article class="remark remark-like" id="remark-1"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.1.2</span><span class="period">.</span>
</h6>
<p id="p-3">In previous classes you likely used the variables \(x,y,z\) in equations. However, since this course often deals with equations of four or more variables, we will often write our variables as \(x_i\text{,}\) and assume \(x=x_1,y=x_2,z=x_3,w=x_4\) when convenient.</p></article><article class="definition definition-like" id="definition-2"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.1.3</span><span class="period">.</span>
</h6>
<p id="p-4">A <dfn class="terminology">system of linear equations</dfn> (or a <dfn class="terminology">linear system</dfn> for short) is a collection of one or more linear equations.</p>
<div class="displaymath">
\begin{alignat*}{5}
a_{11}x_1 &\,+\,& a_{12}x_2 &\,+\,& \dots &\,+\,& a_{1n}x_n &\,=\,& b_1 \\
a_{21}x_1 &\,+\,& a_{22}x_2 &\,+\,& \dots &\,+\,& a_{2n}x_n &\,=\,& b_2\\
\vdots& &\vdots& && &\vdots&&\vdots \\
a_{m1}x_1 &\,+\,& a_{m2}x_2 &\,+\,& \dots &\,+\,& a_{mn}x_n &\,=\,& b_m
\end{alignat*}
</div>
<p id="p-5">Its <dfn class="terminology">solution set</dfn> is given by</p>
<div class="displaymath">
\begin{equation*}
\setBuilder
{
\begin{bmatrix}
s_1 \\
s_2 \\
\vdots \\
s_n
\end{bmatrix}
}{
\begin{bmatrix}
s_1 \\
s_2 \\
\vdots \\
s_n
\end{bmatrix}
\text{is a solution to all equations in the system}
}\text{.}
\end{equation*}
</div></article><article class="remark remark-like" id="remark-2"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.1.4</span><span class="period">.</span>
</h6>
<p id="p-6">When variables in a large linear system are missing, we prefer to write the system in one of the following standard forms:</p>
<div class="sidebyside"><div class="sbsrow" style="margin-left:0.166666666666667%;margin-right:0.166666666666667%;">
<div class="sbspanel" style="width:33.1103678929766%;justify-content:center;">
<p id="p-7">Original linear system:</p>
<div class="displaymath">
\begin{alignat*}{2}
x_1 + 3x_3 &\,=\,& 3\\
3x_1 - 2x_2 + 4x_3 &\,=\,& 0\\
-x_2 + x_3 &\,=\,& -2
\end{alignat*}
</div>
</div>
<div class="sbspanel" style="width:33.1103678929766%;justify-content:center;">
<p id="p-8">Verbose standard form:</p>
<div class="displaymath">
\begin{alignat*}{4}
1x_1 &\,+\,& 0x_2 &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
0x_1 &\,-\,& 1x_2 &\,+\,& 1x_3 &\,=\,& -2
\end{alignat*}
</div>
</div>
<div class="sbspanel" style="width:33.1103678929766%;justify-content:center;">
<p id="p-9">Concise standard form:</p>
<div class="displaymath">
\begin{alignat*}{4}
x_1 & & &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
&\,-\,& x_2 &\,+\,& x_3 &\,=\,& -2
\end{alignat*}
</div>
</div>
</div></div></article><article class="remark remark-like" id="remark-3"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.1.5</span><span class="period">.</span>
</h6>
<p id="p-10">It will often be convenient to think of a system of equations as a vector equation.</p>
<p id="p-11">By applying vector operations and equating components, it is straightforward to see that the vector equation</p>
<div class="displaymath">
\begin{equation*}
x_1 \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}+ x_2 \begin{bmatrix} 0 \\ -2 \\ -1 \end{bmatrix} + x_3 \begin{bmatrix} 3 \\ 4 \\1 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix}
\end{equation*}
</div>
<p data-braille="continuation">is equivalent to the system of equations</p>
<div class="displaymath">
\begin{alignat*}{4}
x_1 & & &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
&\,-\,& x_2 &\,+\,& x_3 &\,=\,& -2
\end{alignat*}
</div></article><article class="definition definition-like" id="definition-3"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.1.6</span><span class="period">.</span>
</h6>
<p id="p-12">A linear system is <dfn class="terminology">consistent</dfn> if its solution set is non-empty (that is, there exists a solution for the system). Otherwise it is <dfn class="terminology">inconsistent</dfn>.</p></article><article class="fact theorem-like" id="fact-1"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">1.1.7</span><span class="period">.</span>
</h6>
<p id="p-13">All linear systems are one of the following:</p>
<ol class="decimal">
<li id="li-1">
<em class="emphasis">Consistent with one solution:</em> its solution set contains a single vector, e.g. \(\setList{\begin{bmatrix}1\\2\\3\end{bmatrix}}\)</li>
<li id="li-2">
<em class="emphasis">Consistent with infinitely-many solutions</em>: its solution set contains infinitely many vectors, e.g. \(\setBuilder
{
\begin{bmatrix}1\\2-3a\\a\end{bmatrix}
}{
a\in\IR
}\)</li>
<li id="li-3">
<em class="emphasis">Inconsistent</em>: its solution set is the empty set \(\{\}=\emptyset\)</li>
</ol></article><article class="activity project-like" id="activity-1"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.1.1</span><span class="period">.</span>
</h6>
<p id="p-14">All inconsistent linear systems contain a logical <dfn class="terminology">contradiction</dfn>. Find a contradiction in this system to show that its solution set is \(\emptyset\text{.}\)</p>
<div class="displaymath">
\begin{align*}
-x_1+2x_2 &= 5\\
2x_1-4x_2 &= 6
\end{align*}
</div></article><article class="activity project-like" id="activity-2"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.1.2</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-1">
<p id="p-15">Consider the following consistent linear system.</p>
<div class="displaymath">
\begin{align*}
-x_1+2x_2 &= -3\\
2x_1-4x_2 &= 6
\end{align*}
</div>
</div>
<article class="task exercise-like" id="task-1"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-16">Find three different solutions for this system.</p></article><article class="task exercise-like" id="task-2"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-17">Let \(x_2=a\) where \(a\) is an arbitrary real number, then find an expression for \(x_1\) in terms of \(a\text{.}\) Use this to write the solution set \(\setBuilder
{
\begin{bmatrix}
\unknown \\
a
\end{bmatrix}
}{
a \in \IR
}\) for the linear system.</p></article></article><article class="activity project-like" id="activity-3"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">1.1.3</span><span class="period">.</span>
</h6>
<p id="p-18">Consider the following linear system.</p>
<div class="displaymath">
\begin{alignat*}{5}
x_1 &\,+\,& 2x_2 &\, \,& &\,-\,& x_4 &\,=\,& 3\\
&\, \,& &\, \,& x_3 &\,+\,& 4x_4 &\,=\,& -2
\end{alignat*}
</div>
<p id="p-19">Describe the solution set</p>
<div class="displaymath">
\begin{equation*}
\setBuilder
{
\begin{bmatrix}
\unknown \\
a \\
\unknown \\
b
\end{bmatrix}
}{
a,b \in \IR
}
\end{equation*}
</div>
<p data-braille="continuation">to the linear system by setting \(x_2=a\) and \(x_4=b\text{,}\) and then solving for \(x_1\) and \(x_3\text{.}\)</p></article><article class="observation remark-like" id="observation-1"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">1.1.8</span><span class="period">.</span>
</h6>
<p id="p-20">Solving linear systems of two variables by graphing or substitution is reasonable for two-variable systems, but these simple techniques won't usually cut it for equations with more than two variables or more than two equations. For example,</p>
<div class="displaymath">
\begin{alignat*}{5}
-2x_1 &\,-\,& 4x_2 &\,+\,& x_3 &\,-\,& 4x_4 &\,=\,& -8\\
x_1 &\,+\,& 2x_2 &\,+\,& 2x_3 &\,+\,& 12x_4 &\,=\,& -1\\
x_1 &\,+\,& 2x_2 &\,+\,& x_3 &\,+\,& 8x_4 &\,=\,& 1
\end{alignat*}
</div>
<p data-braille="continuation">has the exact same solution set as the system in the previous activity, but we'll want to learn new techniques to compute these solutions efficiently.</p></article><article class="remark remark-like" id="remark-4"><h6 class="heading">
<span class="type">Remark</span><span class="space"> </span><span class="codenumber">1.1.9</span><span class="period">.</span>
</h6>
<p id="p-21">The only important information in a linear system are its coefficients and constants.</p>
<div class="sidebyside"><div class="sbsrow" style="margin-left:0.166666666666667%;margin-right:0.166666666666667%;">
<div class="sbspanel" style="width:33.1103678929766%;justify-content:center;">
<p id="p-22">Original linear system:</p>
<div class="displaymath">
\begin{alignat*}{2}
x_1 + 3x_3 &\,=\,& 3\\
3x_1 - 2x_2 + 4x_3 &\,=\,& 0\\
-x_2 + x_3 &\,=\,& -2
\end{alignat*}
</div>
</div>
<div class="sbspanel" style="width:33.1103678929766%;justify-content:center;">
<p id="p-23">Verbose standard form:</p>
<div class="displaymath">
\begin{alignat*}{4}
1x_1 &\,+\,& 0x_2 &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
0x_1 &\,-\,& 1x_2 &\,+\,& 1x_3 &\,=\,& -2
\end{alignat*}
</div>
</div>
<div class="sbspanel" style="width:33.1103678929766%;justify-content:center;">
<p id="p-24">Coefficients/constants:</p>
<div class="displaymath">
\begin{alignat*}{4}
1 & & 0 &\,\,& 3 &\,|\,& 3\\
3 &\, \,& -2 &\,\,& 4 &\,|\,& 0\\
0 &\, \,& -1 &\,\,& 1 &\,|\,& -2
\end{alignat*}
</div>
</div>
</div></div></article><article class="definition definition-like" id="definition-4"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">1.1.10</span><span class="period">.</span>
</h6>
<p id="p-25">A system of \(m\) linear equations with \(n\) variables is often represented by writing its coefficients and constants in an <dfn class="terminology">augmented matrix</dfn>.</p>
<div class="displaymath">
\begin{alignat*}{4}
a_{11}x_1 &\,+\,& a_{12}x_2 &\,+\,& \dots &\,+\,& a_{1n}x_n &\,=\,& b_1\\
a_{21}x_1 &\,+\,& a_{22}x_2 &\,+\,& \dots &\,+\,& a_{2n}x_n &\,=\,& b_2\\
\vdots& &\vdots& && &\vdots&&\vdots\\
a_{m1}x_1 &\,+\,& a_{m2}x_2 &\,+\,& \dots &\,+\,& a_{mn}x_n &\,=\,& b_m
\end{alignat*}
</div>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc|c}
a_{11} & a_{12} & \cdots & a_{1n} & b_1\\
a_{21} & a_{22} & \cdots & a_{2n} & b_2\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn} & b_m
\end{array}\right]
\end{equation*}
</div></article><article class="example example-like" id="example-1"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-example-1"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">1.1.11</span><span class="period">.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-example-1"><article class="example example-like"><p id="p-26">The corresponding augmented matrix for this system is obtained by simply writing the coefficients and constants in matrix form.</p>
<p id="p-27">Linear system:</p>
<div class="displaymath">
\begin{alignat*}{2}
x_1 + 3x_3 &\,=\,& 3\\
3x_1 - 2x_2 + 4x_3 &\,=\,& 0\\
-x_2 + x_3 &\,=\,& -2
\end{alignat*}
</div>
<p id="p-28">Vector equation:</p>
<div class="displaymath">
\begin{equation*}
x_1 \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}+ x_2 \begin{bmatrix} 0 \\ -2 \\ -1 \end{bmatrix} + x_3 \begin{bmatrix} 3 \\ 4 \\1 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix}
\end{equation*}
</div>
<p id="p-29">Augmented matrix:</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c}
1 & 0 & 3 & 3 \\
3 & -2 & 4 & 0 \\
0 & -1 & 1 & -2
\end{array}\right]
\end{equation*}
</div></article></div>
<figure class="figure figure-like" id="figure-1"><div class="video-box" style="width: 100%;padding-top: 56.25%; margin-left: 0%; margin-right: 0%;"><video id="video-E1" class="video" controls=""><source src="https://southalabama.hosted.panopto.com/Panopto/Podcast/Social/b9e50548-7fcd-42a9-a91e-ac190138c4b7.mp4?mediaTargetType=videoPodcast" type="video/mp4"></source>Your browser does not support the <video> tag.<track label="US English" kind="subtitles" srclang="en" src="videos/E1-captions.vtt"></track></video></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">1.1.12<span class="period">.</span></span><span class="space"> </span>Video example: Converting between systems, vector equations, and augmented matrices</figcaption></figure><section class="exercises" id="exercises-1"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">1.1.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-1"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-30">Consider the augmented matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{cccc|c}
1 & -2 & -1 & 3 & 4 \\
2 & -3 & -3 & 5 & 6 \\
0 & -3 & 3 & 3 & 6
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-4">Write a system of scalar equations corresponding to this augmented matrix.</li>
<li id="li-5">Write a vector equation corresponding to this augmented matrix.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-1" id="answer-1"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-1"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-6"><div class="displaymath">
\begin{alignat*}{5}
x_{1} &-& 2 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &=& 4\\
2 \, x_{1} &-& 3 \, x_{2} &-& 3 \, x_{3} &+& 5 \, x_{4} &=& 6\\
&-& 3 \, x_{2} &+& 3 \, x_{3} &+& 3 \, x_{4} &=& 6
\end{alignat*}
</div></li>
<li id="li-7"><div class="displaymath">
\begin{equation*}
x_{1} \left[\begin{array}{c}
1 \\
2 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-2 \\
-3 \\
-3
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-3 \\
3
\end{array}\right] + x_{4} \left[\begin{array}{c}
3 \\
5 \\
3
\end{array}\right] = \left[\begin{array}{c}
4 \\
6 \\
6
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-2"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-31">Consider the vector equation.</p>
<div class="displaymath">
\begin{equation*}
x_{1} \left[\begin{array}{c}
1 \\
1 \\
-1 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-3 \\
-3 \\
3 \\
0
\end{array}\right] + x_{3} \left[\begin{array}{c}
-4 \\
-3 \\
3 \\
-2
\end{array}\right] = \left[\begin{array}{c}
6 \\
4 \\
-4 \\
4
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-8">Write a system of scalar equations corresponding to this vector equation.</li>
<li id="li-9">Write an augmented matrix corresponding to this vector equation.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-2" id="answer-2"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-2"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-10"><div class="displaymath">
\begin{alignat*}{4}
x_{1} &-& 3 \, x_{2} &-& 4 \, x_{3} &=& 6\\
x_{1} &-& 3 \, x_{2} &-& 3 \, x_{3} &=& 4\\
-x_{1} &+& 3 \, x_{2} &+& 3 \, x_{3} &=& -4\\
& & &-& 2 \, x_{3} &=& 4
\end{alignat*}
</div></li>
<li id="li-11"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c}
1 & -3 & -4 & 6 \\
1 & -3 & -3 & 4 \\
-1 & 3 & 3 & -4 \\
0 & 0 & -2 & 4
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-3"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-32">Consider the augmented matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c}
4 & 1 & 7 & 6 \\
-5 & -4 & -6 & -2 \\
-1 & 1 & -3 & -4 \\
-3 & 0 & -6 & -6
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-12">Write a system of scalar equations corresponding to this augmented matrix.</li>
<li id="li-13">Write a vector equation corresponding to this augmented matrix.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-3" id="answer-3"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-3"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-14"><div class="displaymath">
\begin{alignat*}{4}
4 \, x_{1} &+& x_{2} &+& 7 \, x_{3} &=& 6\\
-5 \, x_{1} &-& 4 \, x_{2} &-& 6 \, x_{3} &=& -2\\
-x_{1} &+& x_{2} &-& 3 \, x_{3} &=& -4\\
-3 \, x_{1} & & &-& 6 \, x_{3} &=& -6
\end{alignat*}
</div></li>
<li id="li-15"><div class="displaymath">
\begin{equation*}
x_{1} \left[\begin{array}{c}
4 \\
-5 \\
-1 \\
-3
\end{array}\right] + x_{2} \left[\begin{array}{c}
1 \\
-4 \\
1 \\
0
\end{array}\right] + x_{3} \left[\begin{array}{c}
7 \\
-6 \\
-3 \\
-6
\end{array}\right] = \left[\begin{array}{c}
6 \\
-2 \\
-4 \\
-6
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-4"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-33">Consider the augmented matrix</p>
<div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c}
1 & -1 & -1 & -6 \\
0 & 1 & -3 & 4 \\
0 & 0 & 0 & 1 \\
0 & 2 & -6 & 7
\end{array}\right]
\end{equation*}
</div>
<ol class="lower-alpha">
<li id="li-16">Write a system of scalar equations corresponding to this augmented matrix.</li>
<li id="li-17">Write a vector equation corresponding to this augmented matrix.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-4" id="answer-4"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-4"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-18"><div class="displaymath">
\begin{alignat*}{4}
x_{1} &-& x_{2} &-& x_{3} &=& -6\\
& & x_{2} &-& 3 \, x_{3} &=& 4\\
& & & & 0 &=& 1\\
& & 2 \, x_{2} &-& 6 \, x_{3} &=& 7
\end{alignat*}
</div></li>
<li id="li-19"><div class="displaymath">
\begin{equation*}
x_{1} \left[\begin{array}{c}
1 \\
0 \\
0 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-1 \\
1 \\
0 \\
2
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-3 \\
0 \\
-6
\end{array}\right] = \left[\begin{array}{c}
-6 \\
4 \\
1 \\
7
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><article class="exercise exercise-like" id="exercise-5"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-34">Consider the system of equations</p>
<div class="displaymath">
\begin{alignat*}{4}
-3 \, x &-& y &+& 7 \, z &=& 1\\
& & y &-& 5 \, z &=& 6\\
-x & & &+& 3 \, z &=& 0\\
-2 \, x &-& y &+& 7 \, z &=& -2
\end{alignat*}
</div>
<ol class="lower-alpha">
<li id="li-20">Write an augmented matrix corresponding to this system.</li>
<li id="li-21">Write a vector equation corresponding to this system.</li>
</ol>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-5" id="answer-5"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-5"><div class="answer solution-like"><ol class="lower-alpha">
<li id="li-22"><div class="displaymath">
\begin{equation*}
\left[\begin{array}{ccc|c}
-3 & -1 & 7 & 1 \\
0 & 1 & -5 & 6 \\
-1 & 0 & 3 & 0 \\
-2 & -1 & 7 & -2
\end{array}\right]
\end{equation*}
</div></li>
<li id="li-23"><div class="displaymath">
\begin{equation*}
x_{1} \left[\begin{array}{c}
-3 \\
0 \\
-1 \\
-2
\end{array}\right] + x_{2} \left[\begin{array}{c}
-1 \\
1 \\
0 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
7 \\
-5 \\
3 \\
7
\end{array}\right] = \left[\begin{array}{c}
1 \\
6 \\
0 \\
-2
\end{array}\right]
\end{equation*}
</div></li>
</ol></div></div>
</div></article><p id="p-35"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/E1/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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