16

18

2)

We

Hence, collecting values, we have

r=15a y=žsa

z=5a. may observe that if a is any multiple of 25, the above values of x, y, and 2 will be integers.

(87.) All equations of the first degree, containing any number of unknown quantities, can be solved by either of the Rules under Articles 82, 84, and 86, or by a combination of the same.

The student must exercise his own judgment, as to the choice of the above Rules. In very many cases he will discover many short processes, which depend upon the particular equations given.

(88.) We will now solve a few equations, and shall endeavor to effect their solution in the simplest manner possible.

6x+5y=128) 1. Given to find the values of and

y. Adding the two equations, and dividing the sum by 9, we find x+y=24.

(1) Multiplying (1) by 3, and subtracting the result from the second of the given equations, we have

(2) Subtracting (2) from (1), we get

x=8. S =

x+y=a (1) ? 2. Givenc+z=b (2) to find x, y, and z. y+z=C (3)

3 Dividing the sum of these three equations by 2, we find

atbto x+y+z=

(4) 2

y=16.

From (4) subtracting, successively, (3), (2), and (1), we find

a+b-c
=

2
a-btc
Y

2
-a+b+c

2

and z.

u+x+y=13 (1)

u+x+z=17 (2) 3. Given

to find

U, X, Y, u+y+z=18 (3)

x+y+z=21 (4) Dividing the sum of these four equations by 3, we obtain u+x+y+z=23.

(5) From (5), subtracting successively, (4), (3), (2), and (1), and we find

9

U= 2

2= 5

Y= 66 z=10

x+ay+z)=m (1) 4. Giveny+b(x+x)=n (2) to find x, y, and z.

z+c(x+y)=P (3) If we add and subtract ax from the left-hand member of (1), and add and subtract by from the left-hand member of (2), and add and subtract cz from the left-hand member of (3), they will become (1-a)x+a(x+y+z)=m

(4) (1-bby+6(x+y+z)=n

(5) (1-c)z+c(+y+z)=P

(6) If we divide (4) by 1-a, and (5) by 1-6, and (6) by 1-C, they will become

Taking the sum of (7), (8) and (9), we have

This value of x+y+z substituted in (7), (8) and (9), gives

a

b

(14)

C

5. Given {3x+2y=118 , to find x and y.

x+5y=191

J

added together, the sum is =31; on the other hand, if the first be multiplied by 7, and the second by 4, and both products added together, we shall obtain 68.

Ans. The first is 8, and the second is 3.

8

11. A owes $1200, B $2550; but neither has enough to pay his debts. Lend me, said A to B, of your fortune, and I shall be enabled to pay my debts. B answered, I can discharge my debts, if you will lend me was the fortune of each?

of yours. What

Ans. A's fortune is $900, and that of B $2400.

12. There is a fraction such, that if 1 be added to the numerator, its value, and if 1 be added to the denominator, its value. What fraction is it?

Ans. 15.

13. The sum of two numbers is a, the quotient arising from the division of the one by the other is =b. Find these numbers?

a

ab