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N the solutions of the questions in the
preceding part, the given quantities (being numbers) disappear in the last conclusion, so that no general rules for like cases can be deduced from them. But, if letters are used to denote the known quantities, as well as the unknown, a general solution may be obtained, because, during the whole course of the operation, they retain their original form. Hence also the connection of the quantities will appear in such a manner as to discover the necessary limitations of the data, when there are any, which is essential to the perfe& folution of
a problem. From this method, too, it is
, easy to derive a synthetical demonstration of the solution.
When letters, or any other fuch fymbols, are employed to express all the quantities, the algebra is sometimes called Specious or literal.
E X AMPLE VIII.
To find two numbers, of which the fum and difference are given.
Let s be the given sum, and d the given difference; also, let x and y be the two numbers fought.
Thus, let the given fum be 100, and the difference 24. 1std_124
Thenx=(**!=134-) 62 and 3 (?=)38
In the same manner may the canon be applied to any other values of s and d. By reversing the steps in the operation, it is easy to shew, that if x=
and y =
the sum of x and y must be s, and their
y difference d.
If A and B together can perform a piece of
work in tbe time a, A and C together in the time b, and B and C together in the time c, in what time will each of them perform it alone ?
Let A perform the work in the time x, B in y, and C in %; then, as the work is the fame in all cases, it may be represented by unity.
Mult. 7th by be
9 f =1 and cztey=zy
abc abc IO
abc , abc Mult. 8th by
2abc 2abc. 2abc Add ioth, 11th, 12th 13
actab-bc From 13th subt, twice
bctab-ac From 13th subt. twice '2abc
2abc =bc +ab—ac, & y=
Example in Numbers.
Let a=8 days, b=9 days, and c = 10;
23 then x=1434 y=1721
y=17, and =23 49
31 appears likewise that a, b, c, must be such, that the product of any two of them must be lefs than the sum of these two multiplied by the third.
This is necessary to give positive values of x, y, and
which alone can take place in this queftion. Besides, if x, y, and %, be assumed as any known numbers whatever, and if values of a, b, and c be deduced from steps 7th, 8th, and gth, of the preceding operation, it will appear that a, b, and c, will have the property required in the limitation here mentioned. If a, b, and c were such, that
of the quantities, x, y, or z, became equal
it implies, that one of the agents did nothing in the work. If the values of any of these quantities be negative, the only supposition which could give them any meaning would be, that some of the agents,
instead of pro