7.2 Systems of Linear Equations
A linear equation is a polynomial equation in which each term is of
degree 0 or 1. (The degree of a term of a polynomial is found by adding
the exponents on all the variables of the term.) The following is an
example of a system of linear equations.
Example:
S1
| x + y + 2 z = 5
| y - z = - 2
| z = 3
This particular example is easily solved by the method of back
substitution. The value z = 3 is back substituted into E2 to
yield the equivalent system
S2
| x + y + 2 z = 5
| y = 1
| z = 3
Next, the values z = 3 and y = 1 are back substituted into
E1 to get the equivalent system
S3
| x = -2
| y = 1
| z = 3
for which the solution ( -2, 1, 3 )is apparant.
Exercise 7.2.1
Solve the system by the method of back substitution:
S1
| x + 2 y - z = 3
| y - z = 1
| z = 2
Solution
In each of the systems above, the system is in what is called upper
triangular form. Notice how the left sides of the equations form a
triangular shape. In order to use the method of back substitution, the
system must be in upper triangular form. But what if the system is not in
upper triangular form? If a system is not in upper triangular form, it is
possible to replace it by an equivalent system which is in upper
triangular form by a specific process which we are about to describe.
In order to describe the process of triangularizing a system of linear
equations, it is necessary to return to the subject of what operations on
a system of equations produce equivalent systems.
One of the simplest and most obvious such operations is the operation of
simply switching two equations in the system. For example, if one is
given the following system
S1
| x + 2 y - z = 3
| z = 2
| y - z = 1
The operation E2 <-> E3 merely switches equations two and three of the
system to produce an upper triangular system:
S2
| x + 2 y - z = 3
| y - z = 1
| z = 2
We can symbolize the Switching Principle by saying that operations
of the form En <-> Em always produce equivalent systems.
In the last section, we noted the principle that any equation in a system
can be replaced by an equivalent system. From this principle, we can
deduce a more specific principle that any equation can be replaced by a
non-zero multiple of itself. This principle can be symbolized by saying
that operations of the form En
--> c En where c ≠
0 produce equivalent systems.
The final of the three principles which we will need is more abstract and
less obvious than the first two. It states that any equation in the
system may be replaced by the sum of itself and some multiple of another
equation in the system. This may be symbolized by saying that operations
of the form En --> c
Em + En produce equivalent systems.
Summarizing, the following operations produce equivalent systems:
(1) En <-> Em
(2) En --> c En
(3) En --> c Em + En
Exercise 7.2.2
Perform the following operations on the system:
(1) E2 --> - 2 E1 + E2
(2) E3 --> 3 E1 + E3
S1
| x + 2 y - z = 0
| 2 x + y + z = 0
| - 3 x + y + 2 z = - 6
Solution
Notice that the above operations had the effect of removing the x
terms from E2 and E3, making the system closer to the form of a
triangular system.
Exercise 7.2.3
Perform the operation E2 --> - ( 1 / 3
) E2 on S1
to produce S2.
Then perform the operation E3 --> - 7
E2 + E3 on
S2 to produce a equivalent system S3.
The last system should be upper triangular. Solve it by back
substitution.
S1
| x + 2 y - z = 0
| - 3 y + 3 z = 0
| 7 y - z = - 6
Solution
Exercise 7.2.4
Upper triangularize the system and solve by back substitution.
S1
| x + 2 y - z = 1
| x + z = 3
| y + z = 1
Solution
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