CHAPTER 11
LINEAR EQUATIONS IN ONE VARIABLE
One of the principal reasons for an intensive
study of polynomials, grouping symbols, factoring,
and fractions is to prepare for solving equations.
The equation is perhaps the most important
tool in algebra, and the more skillful the student
becomes in working with equations, the greater will be
his ease in solving problems.
Before learning to solve equations, it is necessary to become familiar with
the words used in the discussion of them. An EQUATION
is a statement that two expressions are equal in
value. Thus,
4+5=9
and
A = lw
(Area of a rectangle = length x width)
are equations. The part to the left of the equality sign is called the LEFT
MEMBER, or first member, of the equation. The part to
the right is the RIGHT MEMBER, or second member, of
the equation.
The members of an equation are sometimes thought of
as corresponding to two weights that balance a scale.
(See fig. 11l.) This compari8on is often helpful to students who are learning
to solve equations. It is obvious, in
Figure 11l. Equation compared to a balance
scale.
the case of the scale, that any change made in one
pan must be accompanied by an equal change in the
other pan. Otherwise the scale will not balance.
Operations on equations are based on the same
principle. The members must be kept balanced or the
equality is lost.
CONSTANTS AND VARIABLES
Expressions in algebra consist of constants and
variables. A CONSTANT is a quantity whose value
remains the same throughout a particular problem. A
VARIABLE is a quantity whose value is free to vary.
There are two kinds of constantsfixed and arbitrary.
Numbers such as 7, 3, 1/2, and p are
examples of FIXED constants. Their values never
change. In 5x + 7 = 0, the numbers 0, 5, and 7, are
fixed constants.
ARBITRARY constants can be assigned different values for different problems.
Arbitrary constants are indicated by lettersquite
often letters at the beginning of the alphabet such as
a, b, c, and d. In
ax+b=0,.
the letters a and b represent arbitrary constants. The form ax t b  0
represent many linear equations. If we give a and b
particular values, say a  5 and b = 7, then these
constants become fixed, for this particular problem,
and the equation becomes
5x t 7 = 0
A variable may have one value or it may have
many values in a dlscuseion. The letters at the end of
the alphabet, such as x, y, z, and w, usually are used
to represent variables. In 5x + 7,
the letter x is the variable. If x = 1, then
and so on for as many values of x as we desire to
select.
If the expression 5x + 7 is set equal to some particular
number, say 23, then the resulting
equality
5x + 7 = 23
holds true for just one value of x. The value is 6,
since
5(6) + 7 = 23
In an algebraic expression, terms that contain a variable are called VARIABLE
TERMS. Terms that do not contain a variable are
CONSTANT TERMS. The expression 5x + 7 contains one variable term and one
constant term. The variable term is 5x, while 7 is the
constant term. In ax + b, ax is the variable term and
b is the constant term.
A variable term often is designated by naming the variable it contains. In 5x
+ 7, 5x is the xterm. In ax + by, ax is the xterm,
while by is the yterm.
