Page 9 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

1.1 Systems of linear algebraic equations and their solutions constitute one of the
major topics studied in the course known as “linear algebra.” In this first section
INTRODUCTION TO we shall introduce some basic terminology and discuss a method for solving such
SYSTEMS OF LINEAR systems.
EQUATIONS

Linear Equations

Any straight line in the -plane can be represented algebraically by an equation of the form

where , , and b are real constants and and are not both zero. An equation of this form is called a linear equation in the
variables x and y. More generally, we define a linear equation in the n variables , , …, to be one that can be expressed in the
form

where , , …, , and b are real constants. The variables in a linear equation are sometimes called unknowns.

EXAMPLE 1 Linear Equations
The equations

are linear. Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the first
power and do not appear as arguments for trigonometric, logarithmic, or exponential functions. The equations

are not linear. is a sequence of n numbers , , …, such that the equation is
. The set of all solutions of the equation is called its solution set or
A solution of a linear equation
satisfied when we substitute , , …,
sometimes the general solution of the equation.

EXAMPLE 2 Finding a Solution Set

Find the solution set of (a) , and (b) .

Solution (a)

To find solutions of (a), we can assign an arbitrary value to x and solve for y, or choose an arbitrary value for y and solve for x. If
we follow the first approach and assign x an arbitrary value t, we obtain

These formulas describe the solution set in terms of an arbitrary number t, called a parameter. Particular numerical solutions can be