CHAPTER VIII.

Simple Equations.

64. An equation asserts that two expressions are equal, but
we do not usually employ the word equation in so wide a sense.

Thus the statement x+3+x+4=2x+7, which is always
true whatever value x may have, is called an identical equation,
or briefly an identity.

The parts of an equation to the right and left of the sign
of equality are called members or sides of the equation, and
are distinguished as the right side and left side.

65. Certain equations are only true for particular values of
the symbols employed. Thus 3x=6 is only true when x=2,
and is called an equation of condition, or more usually an
equation. Consequently an identity is an equation which is
always true whatever be the values of the symbols involved;
whereas an equation (in the ordinary use of the word) is only
true for particular values of the symbols. In the above example
3x=6, the value 2 is said to satisfy the equation. The object
of the present chapter is to explain how to treat an equation of
the simplest kind in order to discover the value which satisfies it.

66. The letter whose value it is required to find is called
the unknown quantity. The process of finding its value is
called solving the equation. The value so found is called the
root or the solution of the equation.

67. The solution of equations, and the operations subsidiary
to it, form an extremely important part of Mathematics. All
sorts of mathematical problems consist in the indirect determination
of some quantity by means of its relations to other
quantities which are known, and these relations are all expressed
by means of equations. The operation in general of solving a
problem in Mathematics, other than a transformation, is firstly
to express the conditions of the problem by means of one or
more equations, and secondly to solve these equations. For
example, the problem which is expressed by the equation above
given is the very simple question, "What is the number such that
if multiplied by 3, the product is 6?" In the present chapter,
it is the second of these two operations, the solution of an equation,
that is considered.
