Linear Algebra | CHAPTER1 / Solving Linear Systems

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Systems of linear equations are common in science and mathematics. These two examples from high school science \cite{Onan} give a sense of how they arise.

The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances.

Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point), the two balances give this system of two equations.

$\begin{align*} 40h+15c &= 100 \\ 25c &= 50+50h \end{align*}$

The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene $\hbox{C}_7\hbox{H}_8$ and nitric acid $\hbox{H}\hbox{N}{\rm O}_3$ to produce trinitrotoluene $\hbox{C}_7\hbox{H}_5{\rm O}_6\hbox{N}_3$ along with the byproduct water (conditions have to be controlled very well, indeed— trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction

$\begin{equation*} x\,{\rm C}_7{\rm H}_8\ +\ y\,{\rm H}{\rm N}{\rm O}_3 \quad\longrightarrow\quad z\,{\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ w\,{\rm H}_2{\rm O} \tag*{}\end{equation*}$

must equal the number present afterward. Applying that principle to the elements C, H, N, and O in turn gives this system.

$\begin{align*} 7x &= 7z \\ 8x +1y &= 5z+2w \\ 1y &= 3z \\ 3y &= 6z+1w \end{align*}$

To finish each of these examples requires solving a system of equations. In each, the equations involve only the first power of the variables. This chapter shows how to solve any such system.

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