Glossary

# Solution of a linear system

Solution of a linear equation

Definition: A solution of a linear equation like $a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$ is any ordered list $(s_1, s_2,\cdots, s_n)$ of $n$ numbers that makes the above equation true when we replace $x_1, x_2,\cdots, x_n$ by $s_1, s_2,\cdots, s_n$, respectively .
Example: $(1, -2, 0)$ is a solution of the equation

$3x_1 + x_2 -\frac{x_3}{3} = 1$

Observe that a linear equation on two variables or more have infinite solutions.

Solution of a Linear system

Definition: Solution of a linear system is any ordered list that is a solution of all equations in the linear system.

Example: The list (or the point) $(3,2)$ is a solution of the following linear system on two variables

$x_1 - 2x_2 = -1$

$-x_1 +3x_2 = 3$

Number of solutions

A linear system has

1. No solution, or
2. Exactly one solution, or
3. Infinitely many solutions.

Question: Why couldn’t there be only two (or three, or four, …) solutions?