4.2.1 Introduction

At first we will restrict ourselves to systems of linear equations in two variables.

Because of their linearity, each of the two equations of the system in info box 4.2.1 can be interpreted as the equation of a line in the $x$-$y$-plane. If, for example, the first equation is solved for $y$,

${\displaystyle y=-\frac{a_{11}}{a_{12}}x+\frac{b_1}{a_{12}}\hspace{0.5em},}$

one immediately sees from this explicit form that it describes a line with the slope $m=-a_{11}/a_{12}$ and the $y$-intercept $y_0=b_1/a_{12}$.
Note that solving the equation for $y$ is only allowed if $a_{12}\ne 0$. For $a_{12}=0$, the first equation reads $a_{11}·x=b_1$; for $a_{11}\ne 0$ this is equivalent to $x=(b_1/a_{11})$, i.e. $x$ is a constant. This equation also describes a line, namely a line parallel to the $y$-axis with $y$-intercept $(b_1/a_{11})$.
And what about the case in which both $a_{12}=0$ and $a_{11}=0$? Then, we also have $b_1=0$, since otherwise the first equation would result in a contradiction. But for $a_{11}=a_{12}=b_1=0$, the first equation is (for all values of $x$ and $y$) always identically satisfied ($0=0$), and hence useless.
The case of the second equation in info box 4.2.1 is similar:

${\displaystyle y=-\frac{a_{21}}{a_{22}}x+\frac{b_2}{a_{22}}\hspace{0.5em}.}$

Altogether, one obtains two lines representing the two linear equations. The question for the solvability and for the solution of the system of linear equations, namely the question for the simultaneous validity of the two equations reads as the question for the existence and the position of the intersection point of the two lines. To this, let us investigate a specific example.

This intuitive approach is very well suited to discussing all cases which can generally occur: two lines in the $x$-$y$-plane can intersect each other - and then the intersection point is necessarily unique -, or the two lines are parallel and thus do not have any intersection point, or the two lines coincide - so to speak - they intersect in an infinite number of points. There are no other cases.
Accordingly, the corresponding system of linear equations has one of the following solution sets:

This will be illustrated by two further examples starting directly with the systems of linear equations: For the example on the right, other parametrisations of the solution set are possible and allowed. It basically just depends on how to describe the points of the (congruent) lines appropriately. The above description of the solution set $L$ simply used the equation of the line itself and the control variable was denoted by $t$ instead of $x$.
And what about the above mentioned restrictions due to the base set? Let us look at the following example.