Chapter 4 System of Linear Equations

Section 4.1 What are Systems of Linear Equations?

4.1.2 Contents

Before we can really start, let us clarify the terminology.

The two equations in the first example 4.1.1 form a system of linear equations in the variables $x$ and $y$. In contrast, the three equations

${\displaystyle x+y+z=3,\mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}}x+y-z=1,\mathrm{\hspace{0.5em}\hspace{0.5em}}\text{and}\mathrm{\hspace{0.5em}\hspace{0.5em}}x·y+z=2}$

do form a system of equations in the variables $x$, $y$, and $z$, but the system is not linear since in the third equation the term $x·y$ occurs, which is bilinear in the variables $x$ and $y$ and hence violates the condition of linearity.
By the way, in a system of equations the number of equations need not be equal to the number of variables; we will return to this later on.
Systems of linear equations are distinguished from general systems of equations by their relative simplicity. Nevertheless, they play an important role in fields as diverse as medical science (e.g. for CT scans), engineering (e.g. for describing sound propagation in complex designed spaces), or physics (e.g. concerning the question of which wave lengths excited atoms can emit). It is, without doubt, worth dealing with systems of linear equations intensely.
For systems of equations generally, the question focuses on which values the variables must take such that all equations of the system are simultaneously satisfied. Such a set of values for the variables is called a solution of a system of equations.
Before we solve systems of equations a detail should be noted: depending on the problem, it may not be useful to accept all variable values. In the first example 4.1.1 the variables $x$ and $y$ are the numbers of unicycles and bicycles the group of stuntmen owns. Such numbers can only be non-negative integers, i.e. elements of $\mathbb{N}{}_0$. Hence, in this case the number range for the solutions has to be restricted to $\mathbb{N}{}_0$ in advance (namely, for both $x$ and $y$).
If the base set is not explicitly specified - and cannot be derived from the problem description - we will assume implicitly that the base set is $ℝ$, the number range of the real numbers.