Dave4Math / Courses / Linear Equations

Linear Equations

Master linear equations. Learn all about matrices, their rank, Gaussian elimination, and the theory of solving equations. With this course, you'll understand when linear systems are solvable and learn strategies on how to find their solutions.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Not Enrolled
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Have you ever had a teacher that was a great communicator? You just felt like they could get anything across to you, and what they were saying surprisingly connected with you. Introducing Dave’s online math course: Linear Equations. Dave knows linear algebra very well and is good at communicating in a very understandable way.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in linear equations.
  • Students who are enrolled in linear algebra and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about matrices, vectors, Gauss–Jordan elimination and to understand the number of solutions of a linear system.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is recommended.

What You’ll Learn in Linear Equations

  • Introduction to Linear Systems
  • Geometric Interpretation
  • A System with Infinitely Many Solutions
  • A System without Solutions
  • Matrices
  • Vectors
  • Gauss–Jordan Elimination
  • The Number of Solutions of a Linear System
  • Matrix Algebra

Course Description

We begin this course with an introduction to linear systems of equations. We start with 2 x 2 systems in two dimensions and 3 x 3 systems in three dimensions. The idea is to get across how many solutions a linear system can have by considering the linear system geometrically. Then we analyze n x m linear systems of equations.

Next, we discuss the matrix form of a linear system and how the solution set to a linear system is written in terms of vectors. We then consider vectors, matrices, matrix operations, and various properties of vectors and matrices.

We examine in great detail techniques on solving linear systems of equations. We also talk about the row-echelon form of a matrix and using Gaussian elimination and Gaussian-Jordan elimination to obtain a row-echelon form and the reduced row-echelon form, respectively.

We also discuss homogeneous systems. Towards the end of the course, we completely characterize a linear system of equations solutions in terms of the system’s rank. We called this the Fundamental Theorem of Linear Systems.

If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on linear algebra.

David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.