2 Equations, 2 Unknowns

Introduction

Consider the below system of equations:

2x - y = 0
-x + 2y = 3

This can be envisioned in three ways.

Row Picture

The equations can be plotted together, and all intersections of the lines are solutions to the system.

row-picture.png

Column Picture

The system is equivalent to the following linear combination of columns.

 ┌  ┐    ┌  ┐   ┌  ┐
 │ 2│    │-1│   │ 0│
x│-1│ + y│ 2│ = │ 3│
 └  ┘    └  ┘   └  ┘ 

Column 1 is represented as a vector to (2,-1); column 2 as (-1,2).

column-picture.png

Any combination of these vectors that leads to (0,3) is a solution to the system. For this system, that solution is 1 of column 1 and 2 of column 2.

column-picture-solution.png

Matrix Picture

┌     ┐   ┌  ┐   ┌  ┐
│ 2 -1│   │ x│   │ 0│
│-1  2│ + │ y│ = │ 3│
└     ┘   └  ┘   └  ┘ 


CategoryRicottone

LinearAlgebra/2Equations2Unknowns (last edited 2025-03-24 17:32:48 by DominicRicottone)