A system of linear equations has1.

A system of linear equations has
1. no solution, or
2. exactly one solution, or
3. infinitely many solutions.
A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.
Matrix Notation
The essential information of a linear system can be recorded compactly in a rectangular array called a matrix. Given the system
x1 — 2x2 + x3 = 0
2x2 — 8x3 = 8 (3)
—4xi + 5X2 + 9X3 = —9
with the coefficients of each variable aligned in columns, the matrix
2 1 —2 13
0 2 —8
_ —4 5 9 _
is called the coefflcient matrix (or matrix of coefflcients) of the system (3), and
1 —2 1 0
0 2 —8 8
—4 5 9 —9

is called the augmented matrix of the system. (The second row here contains a zero be- cause the second equation could be written as 0 • x1 + 2x2 — 8x3 = 8.) An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.
The size of amatrix tells how many rows and columns ithas. The augmentedmatrix (4) above has 3 rows and 4 columns and is called a 3 X 4 (read “3 by 4”) matrix. If m and n are positive integers, an m X n matrix is a rectangular array of numbers with m rows and n columns. (The number of rows always comes first.) Matrix notation will simplify the calculations in the examples that follow.
Solving a Linear System
This section and the next describe an algorithm, or a systematic procedure, for solving linear systems. The basic strategy is to replace one system with an equivalent system (i.e., one with the same solution set) that is easier to solve.
Roughly speaking, use the x1 term in the first equation of a system to eliminate the x1 terms in the other equations. Then use the x2 term in the second equation to eliminate the x2 terms in the other equations, and so on, until you finally obtain a very simple equivalent system of equations.
Three basic operations are used to simplify a linear system: Replace one equation by the sum of itself and a multiple of another equation, interchange two equations, and multiply all the terms in an equation by a nonzero constant. After the first example, you will see why these three operations do not change the solution set of the system.