Overview
On this page, we discuss Systems of Linear Equations (SLE) and what it means to “solve” them. We discuss how to write a SLE in matrix form, and how to apply elementary row operations to simplify it, without changing the set of solutions.
Important
The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class  but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch  you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
 Determine whether a given equation or a given system is linear.
 Determine whether a given \(n\)tuple is a solution of a linear system.
 Find the augmented matrix of a linear system.
 Find the linear system corresponding to a given augmented matrix.
 List the elementary row operations.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
 Find the solution set of an equation.
 Perform elementary row operations on a linear system and on its corresponding augmented matrix.
 Determine whether a linear system is consistent or inconsistent.
 Find the set of solutions to a consistent linear system.
To prepare for class

Watch the following video which shows how a system of 2 linear equations in 2 variables can be solved graphically:

Watch the following video which shows the different types of solution sets when studying the intersection of 2 lines in \(\mathbb{R}^2\) or 3 planes in \(\mathbb{R}^3\):

Watch the following video which shows how a system of 2 linear equations in 3 variables can be solved using basic algebra (which we will avoid in the future):

Watch the following video which shows how to find parametric equations for the infinite set of solutions when 3 planes intersect in one line:

Watch the following video which introduces the concept of a matrix (and important related terms, like the format, entries, and diagonal or triangular matrices), and in particular the augmented matrix of a system of linear equations:

Watch the following video which introduces the 3 elementary row operations which we can use to simplify the augmented matrix of a system of linear equations: