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?

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