Basic Variables, Free Variables, and Parameters
Basic Variables and Free Variables
Suppose the system of linear equations has Am,n+1 for its augmented
matrix. Set R to be the reduced row echelon form of A. Then:
• There are m linear equations in n variables
• If r is the number of pivots in R, then:
• Each pivot represents a basic variable
• The n r non-pivot variables are free variables. Each free variable
can take any value; think of them as parameters in the solution.
Example
Suppose
R=

1 1 0
0 0 1
3 4
2 5
• Then r1,1 and r2,3 are the pivots, so x1 and x3 are the basic variables.
• There are 4 variables
R represents the system
⇢
2 pivots = 2 free variables, x2 and x4 .
⇢
x1 + x2
+ 3x4 = 4
x1 = 4 x2 3x4
=)
x3 + 2x4 = 5
x3 = 5 2x4
The solution to the system is
(x1 , x2 , x3 , x4 ) = (4
x2
3x4 , x2 , 5
2x4 , x4 )
where x2 and x4 are any real numbers.1
1 Sometimes
the parameters.
written as (x1 , x2 , x3 , x4 ) = (4
s
3t, s, 5
2t, t) for s,t 2 R to highlight