4. Given +22 = 18, and x+y= 12, to find the

y

values of x and y.

518

33

Here, we substitute for the unknown quantities, the sum and difference of two auxiliary quantities, as a convenient device to facilitate the solution.

5. Given + y3 = 35, and x + y = 5, to find the values of x and y.

Ans.

Sx=3, or 2.
y = = 2, or 3.

=

6. Given x2+ y2 = 20, and x2-y2 12, to find the values of x and y.

7. Given x+x2y1+y=19, and x2+xy+y2 = 133,

to find the values of x and y.

Ans.

=

9, or 4. ly= = 4, or 9.

Divide the second equation by the first, and to the

result add the first.

8. Given x+y= 9, and x + y = 3, to find

290. Sometimes one of the given equations, or some combination of the two equations, takes the quadratic form.

9. Given x2+ y2+x+y= 18, and xy=6, to find the values of x and y.

Adding twice the second equation to the first, the sum may be put under the form

(x + y)2 + (x + y) = 30,

from which, by completing the square, we obtain

The signs and being taken in connection in the above answer, the one is to be understood as the reverse of the other. That is, x and y must always take opposite signs.

10. Given x2+3y2 — x − y = 18, and xy+x+y=19, to find the values of x and y.

11. Given 2+ 3 x + y = 73 -2xy,

= 44, to find the values of x and y.

Ans.{

x= 4, or 16, or
y=5, or -7, or

4 -11. F√

and y2 + 3y + x

12 ± √58. 158.

291. The following equations may be solved by such of the preceding methods as may be deemed most convenient, or by such devices as skill and ingenuity may suggest.

1. A says to B, "The sum of our money is $18"; B replies, "But if twice the number of your dollars were multiplied by mine, the product would be $154." How many dollars had each ?

Ans. A, $7; B, $11.

2. The difference of two numbers is 5, and the sum of their squares is 193. What are those numbers ?

Ans. 12 and 7.

3. A and B have each a small field, in the shape of an exact square, and it requires 200 rods of fence to enclose both. The contents of these fields are 1300 square rods. What is the value of each, at $2.25 per square rod? Ans. One, $900; other, $ 2025.

4. There are two numbers whose product is 77, and the difference of whose squares is to the square of their difference as 9 to 2. Required the numbers.

Ans. 11 and 7.

5. Two gentlemen, A and B, were speaking of their ages. A said that the product of their ages was 750; B replied, that if his age were increased 7 years, and A's were lessened 2 years, their product would be 851. Required their Ans. A's, 25 years; B's, 30 years.

ages.

6. John Smith's garden is a rectangle, and contains 15,000 square yards, exclusive of a walk, 7 yards wide, which surrounds it, and contains 3696 square yards. Required the length and breadth of the garden.

Ans. Length, 150 yards; breadth, 100 yards.

7. What two numbers are those whose difference multiplied by the less produces 42, and by their sum 133? Ans. 13 and 6.

8. A man bought 10 ducks and 12 turkeys for $22.50. He bought 4 more ducks for $6 than turkeys for $5. What was the price of each ?

Ans. Of a duck, $0.75; of a turkey, $1.25.

9. A man purchased a farm in the form of a rectangle, whose length was four times its breadth. It cost as many dollars per acre as the field was rods in length, and the number of dollars paid for the farm was four times the