Introduction to systems of linear equations
These slides are based on Section 1 in
Linear Algebra and its Applications
by David C. Lay.
Definition 1.
A
linear equation
in the variables
x
1
,...,x
n
is an equation that can be
written as
a
1
x
1
+
a
2
x
2
+ +
a
n
x
n
=
b.
Example 2.
Which of the following equations are linear?
•
4
x
1
−
5
x
2
+2=
x
1
•
x
2
=2( 6
√
−
x
1
)+
x
3
•
4
x
1
−
6
x
2
=
x
1
x
2
•
x
2
=2
x
1
√
−
7
Definition 3.
•
A
system of linear equations
(or a
linear system
) is a collection of one or more
linear equations involving the same set of variables, say,
x
1
,x
2
,...,x
n
.
•
A
solution
of a linear system is a list
(
s
1
,s
2
,...,s
n
)
of numbers that makes each
equation in the system true when the values
s
1
,s
2
,...,s
n
are substituted for
x
1
,x
2
,
...,x
n
, respectively.
Example 4. (Two equations in two variables)
In each case, sketch the set of all solutions.
x
1
+
x
2
= 1
−
x
1
+
x
2
= 0
x
1
−
2
x
2
=
−
3
2
x
1
−
4
x
2
= 8
2
x
1
+
x
2
= 1
−
4
x
1
−
2
x
2
=
−
2
Theorem 5.
A linear system has either
•
no solution, or
•
one unique solution, or
•
infinitely many solutions.
Definition 6.
A system is
consistent
if a solution exists.
Armin Straub
[email protected]
1

How to solve systems of linear equations
Strategy: replace system with an equivalent system which is easier to solve
Definition 7.
Linear systems are
equivalent
if they have the same set of solutions.
Example 8.
To solve the first system from the previous example:
x
1
+
x
2
= 1
−
x
1
+
x
2
= 0
R
2
→
R
2+
R
1
x
1
+
x
2
= 1
2
x
2
= 1
Once in this
triangular
form, we find the solutions by
back-substitution
:
x
2
=1/2
,
x
1
=
Example 9.
The same approach works for more complicated systems.
x
1
−
2
x
2
+
x
3
=
0
2
x
2
−
8
x
3
=
8
−
4
x
1
+ 5
x
2
+ 9
x
3
=
−
9
R
3
→
R
3+4
R
1
x
1
−
2
x
2
+
x
3
=
0
2
x
2
−
8
x
3
=
8
−
3
x
2
+
13
x
3
=
−
9
R
3
→
R
3+
3
2
R
2
x
1
−
2
x
2
+
x
3
= 0
2
x
2
−
8
x
3
= 8
x
3
= 3
By back-substitution:
x
3
=3
,
x
2
=
,
x
1
=
It is always a good idea to check our answer. Let us check that
(
29
,
16
,
3)
indeed solves
the original system:
x
1
−
2
x
2
+
x
3
=
0
2
x
2
−
8
x
3
=
8
−
4
x
1
+ 5
x
2
+ 9
x
3
=
−
9
Armin Straub
[email protected]
2

Matrix notation
x
1
−
2
x
2
=
−
1
−
x
1
+ 3
x
2
=
3
bracketleftbigg
bracketrightbigg
(coefficient matrix)
bracketleftbigg
bracketrightbigg
(augmented matrix)
Definition 10.
An
elementary row operation
is one of the following:
•
(replacement)
Add one row to a multiple of another row.
•
(interchange)
Interchange two rows.
•
(scaling)
Multiply all entries in a row by a nonzero constant.