1 Therefore his ftock was L. 1480, which being tried, answers the conditions of the queftion. CASE II. When there are two unknown quantities. Rule. Two independent equations involving the two unknown quantities, must be derived from the question. A value of one of. the unknown quantities must be derived from each of thefe equations: and thefe two values being put equal to each other, a new equation will arife, involving only one unknown quantity, and may therefore be refolved by the preceding rule. Two equations must be deduced from the queftion; for, from one including two unknown quantities, it is plain, a known value of either of them cannot be obtained; more than two equations would be unneceffary, and if any third condition were affumed at pleasure, most probably it would be inconfiftent with the other two, and a queftion containing three fuch conditions. would be abfurd. It It is to be observed, however, that the two conditions, and hence the two equations expreffing them, must be independent, that is, the one must not be deducible from the other by algebraical reasoning; for, otherwise, there would in effect be only one equation under two different forms, from which no folution can be derived. EXAMPLE III. Two perfons, A and B, were talking of their ages; fays A to B, Seven years ago I was just three times as old as you were, and Seven years hence I shall be just twice as old as you will be: I demand their pre The ages of A and B then are 49 and 21, which answer the conditions. The operation might have been a little shortened, by subtracting the 4th from 5th, and thus 14-y+35; and hence y=21. Therefore, (by 6th) x= (39—14) =49. EXAMPLE IV. A gentleman diflributing money among fome poor people, found he wanted 10s, to be able to give 5 s. to each: therefore be gives each 4 s. only, and finds he has 5 s. left.-To find the number of fillings and poor people. If any question fuch as this, in which there are two quantities fought, can be refolved by means of one letter, the solution is in general more fimple than when two are employed. There muft be, however, two independent conditions, one of which is used in the notation of one of the unknown quantities, and the other gives an equation. Let Let the number of poor be I'Z By 2. and 3. Tranfp. 452-10=4x+5 153=15 The number of poor, poor, therefore, is the number of fhillings is (4x+5=) 65, which anfwer the conditions. EXAMPLE V. A courier fets out from a certain place, and travels at the rate of 7 miles in 5 hours; and 8 hours after, another fets out from the fame place, and travels the fame road at the rate of 5 miles in 3 bours I demand how long and how far the first must travel, before he is overtaken by the fe The firft then travelled 50 hours,, the fe cond (8) 42 hours. The miles travelled by each(? 7y5y-40. -) 70. CASE III. When there are three or more unknown quantities. Rule. When there are three unknown quan tities, there must be three independent equations arifing from the queftion; and from each of thefe a value of one of the unknown quantities must be obtained. By comparing these three values, two equations will arife, involving only two un known quantities, which may therefore be refolved by the rule for cafe 2. In like manner, may the rule be extended to fuch questions as contain four or more unknown quantities; and hence it may be inferred, That, when just as many independent equations may be derived from a queftion, as there are unknown quantities in it, thefe quantities may be found by the refolution of equations. |