At its core, linear systems involve a set of linear equations that can be solved. What does it mean to “solve a linear system”? To solve means to find point(s) of intersection between these lines.

The number of potential solutions increase the more lines are introduced into the system. For example, with two lines, there is a potential of one point of intersection (POI). With three, there is a maximum of two POI. Recall that parallel lines are those that will never meet each other if extended infinitely. Thus, parallel lines do not have a solution (POI).

As a POI is a coordinate with an x and y value, finding a solution requires finding both variables. This is why using two linear equations is integral to finding the POI. Without an extra, there lacks a basis to solve both variables (or even one at that). To do so, and assuming we can determine/know both equations, there are two methods we can use.