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IN

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

a problem. From this method, too, it is easy to derive a synthetical demonstration of the solution.

It When letters, or any other fuch symbols, are employed to express all the quantities, the algebra is sometimes called Specious or literal.

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E XAMPLE VIII.

To find two numbers, of which the fum and difference are given.

Let s be the given fum, and d the given difference; also, let x and y be the two numbers fought.

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Thus, let the given fum be 100, and the difference 24. surd

j 2

:

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=

std

s-d, and

y=

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2

2

the sum of m and y must be s, and their
difference d.

EXAMPLE IX.

If A and B together can perform a piece of

work in the 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,
Bin

Y:

and C in %; then, as the work is the same in all cases, it may be represented by unity

K

By

B

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9

= 1 and cztey=zy y

z Mult. 7th by bc

abc abc

IO + xy

g

abc abc Mult. 8th by

ac X2

X

j ab

ahc, abc Mult. 9th by a

12

=ab zy

у

Z

2abc 2abc2abc Add 10th, 11th, 12th 13

+ + =bc+ac+ab From 13th subt. twice 2abc

2abc Ioth

14
=actab-bc, & ZE

actab-bc From 13th subt, twice 2abc

2abc 15.

=bc+ab-ac, & y= IIth

y

bc+ab-ac From 13th subt. twice 2ábc

2abc

=bc+ac-ab, &x= 12th

bctac—ab

X

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16

Example

Example in Numbers.

34, y

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Let a=8 days, b=9 days, and c = 10;

7 then x=14 sy=171, and >=23

It 49

31 appears likewife 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 z, 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 b, and c; will have the property required in the limitation here mentioned.

b, and c were such, that any of the quantities, x, y, or %, became equal to o, it implies, that one of the agents did nothing in the work. If the values of any of these quantities be negatize, the only supposition which could give them any meaning would be, that some of the agents, instead of pro

moting

a,

If a,

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