#### 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.

##### **Info 4.1.2 **

Several equations relating a specific number of variables

**at the same time**form a so-called

**system of equations**. If the variables in every equation occur only linearly, i.e. at most to the power of $1$, and are only multiplied by (constant) numbers, the system is called a

**system of linear equations**, or

**LS**(linear system).

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

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 $x\xb7y$ 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.

##### **Exercise 4.1.4 **

Which of the following systems is a system of linear equations?

$x+y-3z=0$, $2x-3=y$, and $1.5x-z=22+y$, | |

$\mathrm{sin}(x)+\mathrm{cos}(y)=1$ and $x-y=0$, | |

$2z-3y+4x=5$ and $z+y-{x}^{2}=25$. |

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 $y$).

##### **Info 4.1.5 **

The possible number range for the solutions of a system of equations is called the

**base set**of the system. The

**domain**is the subset of the base set for which all the terms of the equations of the system are

**defined**. For systems of linear equations, base set and domain coincide. Finally, the

**solution set**is the subset of the domain which merges the

**solutions**of the system. The solution set is denoted by $L$.