14、Linear Equations, Probability, and Their Applications

Linear Equations, Probability, and Their Applications

1. Linear Equations

1.1 Systems of Linear Equations

A system of linear equations consists of multiple linear equations. For example, the equations (2x + 3y = 5) and (x - 2y = 2) form a system of linear equations. In (\mathbb{R}^2), these equations represent two lines, which can be the same, parallel (no intersection), or intersect at a single point.

We can rewrite the system using subscripted variables as (2x_1 + 3x_2 = 5) and (x_1 - 2x_2 = 2), and further in matrix and vector form:
[
\begin{bmatrix}
2 & 3 \
1 & -2
\end{bmatrix}
\begin{bmatrix}
x_1 \
x_2
\end{bmatrix}
=
\begin{bmatrix}
5 \
2
\end{bmatrix}