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2.5 Nonhomogeneous Systems 65

in (2.5.1). To understand why p appears and where it comes from, consider the
nonhomogeneous system

x 1 +2x 2 +2x 3 +3x 4 =4,
2x 1 +4x 2 + x 3 +3x 4 =5, (2.5.2)
3x 1 +6x 2 + x 3 +4x 4 =7,

in which the coefficient matrix is the same as the coefficient matrix for the
homogeneous system (2.4.1) used in the previous section. If [A|b] is completely
reduced by the Gauss–Jordan procedure to E [A|b]

   
1223 4 1201 2
[A|b]=  2413 5  −→  0011 1  = E [A|b] ,
3614 7 0000 0

then the following reduced system is obtained:

x 1 +2x 2 + x 4 =2,
x 3 + x 4 =1.

Solving for the basic variables, x 1 and x 3 , in terms of the free variables, x 2
and x 4 , produces
x 1 =2 − 2x 2 − x 4 ,
x 2 is “free,”
x 3 =1 − x 4 ,
x 4 is “free.”
The general solution is obtained by writing these statements in the form

2
       −2   −1 
x 1 2 − 2x 2 − x 4
x 2
 .
 x 2     0   1   0  (2.5.3)
1
  =   =   + x 2  0  + x 4  −1
x 3 1 − x 4
0 0 1
x 4 x 4
As the free variables x 2 and x 4 range over all possible numbers, this generates
all possible solutions of the nonhomogeneous system (2.5.2). Notice that the
2
 
column   in (2.5.3) is a particular solution of the nonhomogeneous system
 0 
1
0
(2.5.2)—it is the solution produced when the free variables assume the values
x 2 = 0 and x 4 =0.