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Chapter 1
Linear Equations vs. Nonlinear Equations
Definition
In general, a
linear equation with n unknowns
is an equation that can be placed in the
form a
1
x
1
+ a
2
x
2
+ a
n
x
n
… = b
In the equation above, a have known values and are called coefficients
; b also has a known value
and is called the constant
for the equation, while x are the variables
or unknowns
.
The key characteristic of a linear equation is that the variables appear to the first power only. In
particular, a linear equation never contains products of variables (such as x
1
x
2
), functions of
variables (such as sinx
2
), or powers of variables (such as x
3
5
).
Consistent Systems and Inconsistent Systems
We will frequently refer to a system of m linear equations in n unknowns as an (m x n) linear
system. If m = n (so that there are exactly as many equations as there are unknowns) we will call
the system a square system. If a system is not square
, it is called a rectangular
system.
Definition
If an (m x n) system has at least one solution, then we say the system is
consistent
. If
the system has no solutions whatsoever, then it is called
inconsistent
.
Elementary Operations in 2 x 2 Systems
This module introduces Gauss-Jordan elimination
, a practical and systematic method for solving
systems of linear equations.
We will restrict ourselves in this module to square systems that have unique solutions. Later, in
Module 1.2 and 1.3, we will treat rectangular systems and systems that have no solutions or
infinitely many solutions.
The solution procedure we want to discuss, Gauss-Jordan elimination, is based on an idea that is
probably familiar to you--that of using elementary operations to eliminate some of the variables
from an equation.
Elementary Operations
1. Interchange any two equations (
)
2. Multiply both sides of any equation by a nonzero constant (
)

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3. Replace any equation by adding a constant multiple of any other equation to it (
)
It can be shown that if we use any of the three elementary operations to create a new system,
then the new system has exactly the same solutions as the original system. We call two systems
having the same solution set
equivalent systems
. Thus, whenever we apply an elementary
operation, we produce a system equivalent to the one we started with.
The Solution Possibilities for a 2 x 2 system
Geometric representations of (2 x 2) linear systems
The graph of a linear equation in two unknowns is a line in the plane. Thus, a system of two
linear equations in two unknowns can be represented geometrically by two lines in the plane.
The solution to the system is found at the intersection of the two lines.

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