SECT. II. Solution of Questions producing Simple Equations.

From the resolution of equations we obtain the refolution of a variety of useful problems, both in pure mathematics and physics, and also in the practical arts founded upon these sciences. In this place, we confider the application of it to those queftions where the quantities are expreffed by numbers, and their magnitude alone is to be confidered.

When an equation, containing only one unknown quantity, is deduced from the queftion by the following rules, it is fometimes called a Final Equation. If it be fimple, it may be refolved by the preceding rules; but, if it be of a superior order, it must be refolved by the rules afterwards to be explained. The examples in this chapter are so contrived, that the final equation may be fimple.

The rules given in this section, for the folution of questions, though they contain

a

a reference to fimple equations only, are to be confidered as general, and as applicable to questions which produce equations of any order.

GENERAL RULE.

The unknown quantities in the question propofed must be expressed by letters, and the relations of the known and unknown quantities contained in it, or the conditions of it, as they are called, must be expreffed by equations. Thefe equations being refolved by the rules of this fcience, will give the answer of the question.

For example, if the queftion is concerning two numbers, they may be called x and y, and the conditions from which they are' to be investigated, must be expreffible by equations.

Thus, if it be required that the fum of two numbers fought be 60, that condition. is expreffed thus, x+y=60.

If their difference must be 24,

then

If their product is 1640, then,

xy=24 xy=1640.

If their quotient must be 6, then =6.

If their ratio is as 3 to 2, then xy:: 3:2, and therefore

y

These are fome of the relations, which are most easily expreffed: Many others occur, which are lefs obvious; but, as they cannot be described in particular rules, the algebraical expreffion of them is beft explained by examples, and must be acquired by experience.

A diftinct conception of the nature of the queftion, and of the relations of the feveral quantities to which it refers, will generally lead to the proper method of ftating it, which in effect may be confidered only as a tranflation from common language, into that of Algebra.

CASE I. When there is only one unknown quantity to be found.

Rule.

Rule. An equation involving the unknown quantity must be deduced from the question (by the general rule). This equation being refolved by the rules of the laft fection, will give the answer.

It is obvious, that, when there is only one unknown quantity, there must be only one independent equation contained in the queftion; for any other would be unneceffary, and might be contradictory to the former.

EXAMPLE I

To find a number, to which if there be added a half, a third part, and a fourth part of itself, the fum will be 50.

Let it be z: then half of it is,

a third

If the operation be more complicated, it may be useful to register the several steps of it, as in the following

EXAMPLE II.

A trader allows L. 100 per annum for the
expences of his family, and augments year-
ly that part of his flock which is not fo
expended, by a third part of it; at the
end of three years, his original flock was
doubled: What had be at first?

Let his first stock be

Of which he fpends the first year fool. and there remains This remainder is increased by a third of itfelf

The fecond year he fpends 100l. and there remains

He increases the re-
mainder by one third
of it.

The third year he
fpends 100 l. and
there remains
He increases it by
one third.

But at the end of the'
third year his stock
is doubled; therefore,

By R. 3.
By R. 1.
By R. 2.