Chapter 4 System of Linear Equations

Section 4.1 What are Systems of Linear Equations?

4.1.1 Introduction

A problem in which several variables occur at the same time!? And on top of that, a whole slew of equations is involved!? Problems of this kind do not only occur in science and engineering but also in other academic fields and in every day life! And they all have to be solved!
But calm down now: it won't get difficult! However, it is true that in very diverse fields you will often encounter situations and problems which can be mathematically modelled by several equations in several variables. Here, we consider a simple first example.
Example 4.1.1
A young group of stuntmen want to pimp up their breakneck cycling stunt by purchasing new rims which create garish light effects for their unicycles and bicycles. For a total of $10$ unicycles and bicycles $13$ rims are required. How many unicycles and bicycles do the group have?
The first step is to translate the information given in the description of the problem into mathematical equations. Let $x$ denote the desired number of unicycles and $y$ denote the number of bicycles. Then the first information given in the problem reads

$\begin{array}{ccc}\multicolumn{1}{c}{\text{equation} \left(1\right)}& :\hfill & x+y=10\hfill \end{array}$

since the group have $10$ cycles in total. Moreover, a unicycle has one rim and a bicycle has two rims. Since $13$ rims are to be purchased in total, it is also known that

$\begin{array}{ccc}\multicolumn{1}{c}{\text{equation} \left(2\right)}& :\hfill & x+2y=13.\hfill \end{array}$

Thus, from the problem description two equations arise relating the two variables $x$ (number of unicycles) and $y$ (number of bicycles).

Of course, sooner or later you want to know how many unicycles and bicycles the group of stuntmen really has. In the given example you can guess the values of $x$ and $y$ by a little trial and error. Here we are actually interested in the methods for solving problems like the example above systematically.