My current lesson is the second in the introduction to systems of linear equations and matrices. I understand how to use matrices to solve a system of linear equations and I understand how to add and multiply matrices.

I don't understand why one would want to perform these functions. I tried graphing two systems of linear equations, identifying their solutions (intersection points). Then I added the associated matrices and graphed the resulting matrix. The solution of the result doesn't seem to have any meaningful relationship to the two original matrices nor their solutions.

I understand the application of a single matrix to demonstrate Gaussian elimination as in the example:

A football game has a total score of 39 points consisting of 3-point field goals, 6-point touchdowns and 1-point extra points. There were 11 plays in the game. There were the same number of touchdowns as field goals. How many of each type of score were there?

But why then, would anyone multiply this system of linear equations by some other system? What could one hope to gain?

I know this sounds esoteric but the next lesson is Inverses and Determinants of Matrices. I'd like to have an idea of the bigger picture before delving into the details.