Section 6 Examples

Chemical Equations (YAY!)

Given the combustion reaction $x_1CH_4 + x_2O_2 \rightarrow x_3CO_2 +x_4H_2O$, what are the values of $\vec{x}$?

Using Linear Algebra and assigning each element to a ROW we can redefine this apparently difficult problem to a simple matrix

\begin{align} \begin{matrix} C\\H\\O \end{matrix} \begin{bmatrix}1&0&-1&0&|&0\\4&0&0&-2&|&0\\0&2&-2&-1&|&0 \end{bmatrix} = \begin{bmatrix} 1&0&-1&0&|&0\\0&0&4&-2&|&0\\0&4&0&-2&|&0 \end{bmatrix} \end{align}

Therefore, the $\vec{x} = x_3 \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix} + x_4 \begin{bmatrix} 0\\1\\\frac{1}{2}\\1 \end{bmatrix}$. Now, chemists are afraid of fractions, so we choose an $x_4$ such that it's only whole numbers, so why not $x_4 = 2$.

Network Flow

I apologize that no picture will be accompanying this one because I have no idea how to make a diagram of that complexity in LaTeX and I don't really feel like learning all that right now…so! Here's the situation
Note: All positive integers are flow in, all negative integers are flow out.

At point A
$x_5 +x_1 -300 +x_2 = 0$
$x_2 + 200 -700 - x_3 = 0$
$x_3 + x_4 -200 + x_1 = 0$
$x_5 + x_4 + 200 = 200 + 300 + 700$

\begin{align} \begin{bmatrix} 1&-1&0&0&1&|&300\\0&1&-1&0&0&|&700\\-1&0&1&1&0&|&200\\0&0&0&1&1&|&1000 \end{bmatrix} = \begin{bmatrix} 1&0&1&0&1&|&800\\0&1&1&0&0&|&500\\0&0&0&1&1&|&1000\\0&0&0&0&0&|&0 \end{bmatrix}. \end{align}
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