Solving Systems of Equations

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This quarter I have stumbled into a rather interesting situation: I am tutoring a class that is learning exactly what my Linear Algebra class is going over. Specifically, this is biting me in the ass because of our focus on Solving Systems of Linear Equations. In order to help my tutorial groups, as well as jogging my own mind with all the insanity of Algebra 2, I am going to walk through three important, and rather involved, methods of solving them.

The three routes we will be using are Back Substitution, Gaussian Elimination and finally, the most well anticipated and least appreciated – Cramer’s Determinant Rule. Each of the three have pros and cons for when to use them and why not to so I will break each into its own post so I have room and time to actually address these aspects and avoid posting a huge ugly post that no one will actually read.

For the next short while I would like to define the three methods, introduce our system and find a solution using Back Substitution.

Our Methods

Back Substitution is one of the more intuitive methods for solving systems of equations. We will work through the system together to illustrate it in all of its glamour. The method is pretty simple and done in 6 easy steps… which beats the 12 step program some of us are familiar with.

  1. Choose one of the equations
  2. Choose one of the variables to solve for
  3. Solve for it
  4. Substitute that back into another equation, eliminating one of the variables
  5. Repeat again until there are as few variables as possible (hopefully only one)
  6. Evaluate and step backwards through the equations to solve for the other values.

I note in #5 that you should be aiming for the fewest variables because of the possible occurrence when we are not given enough equations, but i digress. Suffice to say that once you see it in action you will understand.

Although not as intuitive as Back Substitution, Gaussian Elimination is my favorite method. It makes use of a couple other tools, like matrices and row manipulation and the Reduced Row Echelon Form. Gauss made some really beautiful influential pressure on the world of mathematics, and Wilhelm Jordan came through shortly before Gauss died to make further improvements. The history isn’t super important, except that it makes you sound pompous and arrogant, which can be pretty useful if you want to get out of a date with a girl who has said Like too many times in the last ten minutes.

Finally, and certainly least, Cramer’s Determinant Rule. This is one of the more irritating methods because of the choice of turning to the determinant. Everyone and their mother decides to teach this a totally different way depending on the size of the matrix and their preference for rows or columns. It is incredibly difficult to explain that the methods all end in the same result when the students are already bugging out because of the determinant in the first place. Rest assured they are all the same and if my method is not identical to your instructors… you are welcome to tell your instructor to come and correct me.

Our System of Equations

For the purposes of this series of articles we will be using the same system of equations.





You should see above that we have three variables (x, y, z) and four equations. This is typically a sign that something is off. For the most part when you are given an equation you are able to solve for an unknown, also known as a variable. For those of you who like neat little sayings you may want to get this tattooed across your shoulders:

With N equations, N unknowns

Dealing with this pearl of wisdom within the bounds of our system described above should leave a very good question in your mouth – Why do we have four equations and three unknowns? The truth is, we don’t, but I’ll let discussion about this come out as time goes on.

These equations are also able to be referred to as limiting factors or influencing factors depending on the instance and person talking. Personally I will try to keep things simple and always refer to them as a system of equations.

Back Substitution Time

Since this is not an exercise in algebraic manipulation ill go ahead and show you the steps I would take. Make sure you note that there are a million routes here as well. You don’t have to follow my road.









You can take this point and plug this point into one of the equations above and you will get a beautiful check mark on your paper. Since we have a solution to the equation this is our checkmark for all of the following solutions and we should be able to say that we understand how to solve systems of equations with confidence.