14. What fraction is that, from the numerator of which if 3 be subtracted, its value will be, but if 3 be subtracted from its denominator its value will be?

The reason why numbers chosen indiscriminately will not satisfy the conditions of the above formulas will be explained hereafter.

Equations with several Unknown Quantities.

XXIII. Questions involving more than two unknown Quan

tities.

Sometimes it is necessary to employ, in the solution of a question, inore than two unknown quantities. In this case, the question must furnish conditions enough to form as many distinct equations as there are unknown quantities.

1. A market woman sold to one man, 7 apples, 10 pears, and 12 peaches, for 63 cents; and to another, 13 apples, 6 pears, and 2 peaches, for 31 cents; and to a third, 11 apples, 14 pears, and 8 peaches for 63 cents. She sold them each time at the same rate. What was the price of each?

The second being multiplied by 6, the z will have the same coefficient as in the first.

If the second be multiplied by 4, the z will have the same coefficient as the 3d.

71x+26 y

123

41 x + 10 y

61

We have now the two equations

and which contain only two unknown quantities. These may now be reduced in the same manner as others with two unknown quantities.

• Multiplying the 5th by 5, and the 7th by 13, the coefficient of y will be the same in both.

We have now found an equation containing only one unknown quantity.

178 x 178

x= 1.

Putting the value of x into the 7th, it becomes

41 + 10 y = 61

Ꮖ

10 y = 20

y= 2.

Putting the values of x and y into the 2d, it becomes

13 + 12+2 z = 31

22= 6
z= 3.

Ans. The apples 1, the pears 2. and the peaches 3 cents each.

In the same manner, questions, involving four unknown quantities, may be solved. First combine them two by two till one of the unknown quantities is eliminated from the whole, and there will be three equations with three unknown quantities. Then combine these three two by two, until one of the un

known quantities is eliminated, and then there will be two equations with two unknown quantities, and so on.

Either of the methods of elimination may be used as is most convenient.

It is not necessary that all the unknown quantities should enter into every equation.

2. A market woman sold at one time 7 eggs, 12 apples, and a pie for 26 cents; at another time 12 eggs, 18 pears, and 3 pies, for 69 cents; at a third time 20 pears, 10 apples, and 17 eggs for 69 cents; and at a fourth time, 7 pies, 18 apples, and 10 pears for 66 cents. Each article was sold, at every sale, at the same price as at first. What was the price of each ar ticle ?

Let u the price of an egg,

Putting this value of y into the 2nd and 4th, they become

If the 9th be multiplied by 2, the coefficient of z will be the same as in the 3d;

it the 8th be multiplied by 5, and the 9th by 9, the coefficients of z will be alike.

Deducing the value of x from 11th, and also from 14th.

Making these values of x equal, we have an equation containing only one unknown quantity.

Putting this value of u into the 15th or 16th, we shall find

1

X ==
2

Putting these values of x and u into the 1st, 2nd, or 4th, and we shall find

y = 6.

Putting the values of r and u into the 3d, and we shall find

Ꮖ

2 = 11.

Ans. Eggs, 2 cents each, apples, cent, pears, 19 cent, and pies, 6 cents.

* If the learner is at a loss how to subtract -233 from 69 let him transpose both into the first member, or some terms from the first to the second

In this example, three different methods of elimination were employed. This was not necessary; either method might have been used for the whole. It is sometimes convenient to use one, and sometimes the other.

3. There are three persons, A, B, and C, whose ages are as follows; if B's age be subtracted from A's, the difference will be C's age; if five times B's age and twice C's age be added together, and from their sum A's age be subtracted, the remainder will be 147; the sum of all their ages is 96. What are their ages?

4. Three men, A, B, C, driving their sheep to market, says A to B and C, if each of you will give me 5 of your sheep, I shall have just half as many as both of you will have left. Says B to A and C, if each of you will give me 5 of yours, I shall have just as many as both of you will have left. Says C to A and B, if each of you will give me 5 of yours, I shall have just twice as many as both of you will have left. How many had each?

5. It is required to divide the number 72 into four such parts, that if the first part be increased by 5, the second part diminished by 5, the third part multiplied by 5, and the fourth part divided by 5, the sum, difference, product, and quotient, shall all be equal.

6. A grocer had four kinds of wine, marked A, B, C, and D. He mixed together 7 gallons of A, 5 gallons of B, and 8 gallons of C, and sold the mixture at $1.21 per gallon. He also mixed together 3 gallons of A, 10 of C, and 5 of D, and sold the mixture at $1.50 per gallon. At another time he mixed 8 gallons of A, 10 of B, 10 of C, and 7 of D, and sold the whole for $48. At another time he mixed together 18 gallons of A, and 15 of D, and sold the mixture for $48. What was the value of each kind of wine?

7. Find the values of u, x, y, and z, in the following equations.