System of Linear Equations

Info

These notes were directly imported from Notion. They’re not in Atomic Notes style.

$2x_1-3x_2+5x_3=7$ is a Linear Equation. Multiple would be a SLE. The Solution to this System could not exist, there could be just one solution (ie just one possible value for x1,x2,x3 respectively) or there could be infinitely many.

A system with at least one solution is called consistent, otherwise it’s inconsistent.

Parametric form is just fancy way for saying “in completely different variables”. IE, the parametric form of the original equation could be $2x_1=3v+5s$, $v=x_2$ and $s=x_3$.

When there’s only two variables involved (ie $x+y=2$), the SLE described geometrically. If the system has just one equation, ie the one above, there could be infinitely many solutions. ($1.5+5=2$, $2+0=2$ ).

If there’s two equations in the system, there’s 3 possibilities:

1. The 2 equations intersect at a unique point. IE The system has a single solution.
2. The lines are distinct and parallel. They do not intersect. There’s no solution to this system.
3. The lines are identical. The system has infinitely many solutions for every point they intersect. (Think: $x+y=2$ and $2x+2y=4$).

When trying to solve SLE’s with more than 3 variables, it’s important to use the Matrix Method. This method is outlined in ILA notebook. It’s not easy.