Chapter 1
LINEAR EQUATIONS
1.1
Introduction to linear equations
A
linear equation
in
n
unknowns
x
1
, x
2
,
· · ·
, x
n
is an equation of the form
a
1
x
1
+
a
2
x
2
+
· · ·
+
a
n
x
n
=
b,
where
a
1
, a
2
, . . . , a
n
, b
are given real numbers.
For example, with
x
and
y
instead of
x
1
and
x
2
, the linear equation
2
x
+ 3
y
= 6 describes the line passing through the points (3
,
0) and (0
,
2).
Similarly, with
x, y
and
z
instead of
x
1
, x
2
and
x
3
, the linear equa-
tion 2
x
+ 3
y
+ 4
z
= 12 describes the plane passing through the points
(6
,
0
,
0)
,
(0
,
4
,
0)
,
(0
,
0
,
3).
A
system
of
m
linear equations in
n
unknowns
x
1
, x
2
,
· · ·
, x
n
is a family
of linear equations
a
11
x
1
+
a
12
x
2
+
· · ·
+
a
1
n
x
n
=
b
1
a
21
x
1
+
a
22
x
2
+
· · ·
+
a
2
n
x
n
=
b
2
.
.
.
a
m
1
x
1
+
a
m
2
x
2
+
· · ·
+
a
mn
x
n
=
b
m
.
We wish to determine if such a system has a solution, that is to find
out if there exist numbers
x
1
, x
2
,
· · ·
, x
n
which satisfy each of the equations
simultaneously.
We say that the system is
consistent
if it has a solution.
Otherwise the system is called
inconsistent
.
1