Let the greater number be denoted by x, and the less

If we multiply the first and second equations together,

we obtain,

(3.)

(4.)

= 89

2xy.

Placing this value of x2 - y2, equal to that found above.

and we have,

89

924 2xy= ; xy

or,

89xy 2x2y2 = 924;

and, placing xyz, we obtain,

and hence, by changing the signs and dividing, we have,

Substituting the first value of z for xy, in Equations (3) and (4), gives x = 7, and y = 4; and substituting the second value, z' = 16.5, for xy, in the same equations, we find,

Let the numbers be denoted by 22 and y2.

From the second equation, we have, by transposing,

Substituting this value in Equation (1), we have,

or, if we take the minus sign, then y = 6. If we take y = 8, we find x = 6, and if we take y = 6, we find x = 8; hence, the numbers are 64 and 36.

(14.)

Let the numbers be denoted by x and y.

Substituting this value in the second, we obtain,

If we take the first root, 84, the value of x will be - Co, and these two numbers will satisfy the two equations of condition. But the enunciation of the question required the number 24 to be divided into two parts, and this required that neither x nor y should have a value exceeding 24; hence, we must take the second value of y = 10. This gives x = 14.

(15.)

Let the numbers be denoted by x and y.

By cubing both members of Equation (1), we have,

x3 + 3x2y + 3xy2+ y3 = 512

(3.)

and, by subtracting the second equation from the third, we

we have,

8xy = 120; or, xy = 15.

Combining this with Equation (1), we readily find,

x = 3, and y = 5.

(16.)

Let the number of yards sold by the first, be denoted by x, and the number sold by the second, by y.

Now, if the whole amount received, for any number of things sold, be divided by the number of things, the quo tient will be the cost of each thing. Hence, if 24 dollars be divided by the number of yards of stuff sold by the second, the quotient will be the amount per yard received by the first; and, for a like reason, 12 divided by x, will be the amount per yard received by the second.

But, the first sold 2 yards, and the second y yards; and,

if the amount per yard be multiplied by the number of yards, the product will be the amount received. Hence,

Then, by clearing the first equation of fractions, we have, 24x2 + 124y2 = 35xy;

and, by substituting for y its value, x + 3, we obtain, 24x2 + 12(x2 + 6x + 9)

=

35x(x + 3);

24x2 + 12x2 + 75x + 112 = 35x2 + 105x;

that is,

which gives,

112;

x = 10 ± 5 = 15, and 5;

from which we have the corresponding values of y = or y = 8.

18,

(17.)

Let the highest rate of interest be denoted by y, and the lowest by z. Now, as the incomes are to be equal, it is plain that the first sum put at interest will be the least, which let us denote by x. Then, the larger, or second part will be denoted by 13000 x. Then, since the amount of interest on any sun is equal to the sum, multiplied by the

rate, divided by 100, we have,

by first condition,

У

= (13000