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<h4 class="subsection" id="Solving-Systems-of-Linear-Equations-1"><span>1.2.4 Solving Systems of Linear Equations<a class="copiable-link" href="#Solving-Systems-of-Linear-Equations-1"> ¶</a></span></h4>
<p>Systems of linear equations are ubiquitous in numerical analysis.
To solve the set of linear equations <code class="code">A<var class="var">x</var> = b</code>,
use the left division operator, ‘<samp class="samp">\</samp>’:
</p>
<div class="example">
<pre class="example-preformatted"><var class="var">x</var> = A \ b
</pre></div>
<p>This is conceptually equivalent to
<code class="code">inv (A) * b</code>,
but avoids computing the inverse of a matrix directly.
</p>
<p>If the coefficient matrix is singular, Octave will print a warning
message and compute a minimum norm solution.
</p>
<p>A simple example comes from chemistry and the need to obtain balanced
chemical equations. Consider the burning of hydrogen and oxygen to
produce water.
</p>
<div class="example">
<pre class="example-preformatted">H2 + O2 --> H2O
</pre></div>
<p>The equation above is not accurate. The Law of Conservation of Mass requires
that the number of molecules of each type balance on the left- and right-hand
sides of the equation. Writing the variable overall reaction with
individual equations for hydrogen and oxygen one finds:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">x1*H2 + x2*O2 --> H2O
H: 2*x1 + 0*x2 --> 2
O: 0*x1 + 2*x2 --> 1
</pre></div></div>
<p>The solution in Octave is found in just three steps.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">octave:1> A = [ 2, 0; 0, 2 ];
octave:2> b = [ 2; 1 ];
octave:3> x = A \ b
</pre></div></div>
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