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Systems of Linear Equations

2 min read

Solutions and Solutions Sets

Here are a few examples of linear equations:

  • (we can have 3, 4 basically any amount of variables)

Systems of equations just means multiple equations

We can solve systems of equations using methods like elimination, substitution, etc. There’s also a graphical interpretation for solving:

Equivalent Linear Systems of Equations

AND 

left=-25; right=-5;
top=10; bottom=-5;
---
x+5y=6
2x+7y=-3

At this point, the professor demonstrated, graphically, that 2D systems of linear equations can either have one solution, zero solutions, or infinity solutions

In the same way, they demonstrated that 3D systems of linear equations behave similarly

Just an fyi we can graph up to 3 spatial dimensions, but when we get into higher-order dimensions— we cannot. We, therefore, must rely on the algebra.

Consistent vs Inconsistent Systems

When a system has solutions it is called consistent, else it is called inconsistent.

Augmented Matrix Notation

Let’s solve it via matrices, our , becomes:

This is a 2x3 matrix (in the same way that this would be a 2x3 array) This is called the AUGMENTED MATRIX of the system

Valid Row Operations

  1. Swap Rows
  1. Multi a row by a nonzero constant
  1. Add to a row a multiple of another row
  1. (Special) Replace a row by a combination of a mulitple of that row with a multiple of another row (e.g. )

1/7/25

Graph View