18. A man left $31,500 to be divided equally among his children. But since 3 of the children died, each remaining child received $3375 more. How many children survived?

19. Two bodies move from the vertex of a right angle along its sides at the rate of 12 feet and 16 feet a second respectively. After how many seconds will they be 90 feet apart?

20. A tank can be filled by two pipes, by the one in two hours less time than by the other. If both pipes be open 17 hours, the tank will be filled. How long does it take each pipe to fill the tank?

21. From a thread, whose length is equal to the perimeter of a square, 36 inches are cut off, and the remainder is equal in length to the perimeter of another square whose area is four-ninths of that of the first. What is the length of the

thread?

22. A number of coins can be arranged in a square, each side containing 51 coins. If the same number of coins be arranged in two squares, the side of one square will contain 21 more coins than the side of the other. How many coins does the side of each of the latter squares contain?

23. A farmer wished to receive $ 2.88 for a certain number of eggs. But he broke 6 eggs, and in order to receive the desired amount he increased the price of the remaining eggs by 2 cents a dozen. How many eggs had he originally ?

24. Two bodies move toward each other from A and B respectively, and meet after 35 seconds. If it takes the one 24 seconds longer than the other to move from A to B, how long does it take each one to move that distance?

25. It takes a boat's crew 4 hours and 12 minutes to row 12 miles down a river with the current, and back again against the current. If the speed of the current be 3 miles an hour, at what rate can the crew row in still water?

26. A man paid $300 for a drove of sheep. By selling all but 10 of them at a profit of $2.50 each, he received the amount he paid for all the sheep. How many sheep did he buy?

CHAPTER XIX.

SIMULTANEOUS QUADRATIC AND HIGHER

EQUATIONS.

1. The solution of a system of quadratic or higher equations in general involves the solution of an equation of higher degree than the second, and therefore cannot be effected by the methods for solving quadratic equations. But there are many special systems whose solutions can be made to depend upon the solutions of quadratic equations.

The proofs of the following methods are given in School Algebra, Ch. XXIV.

When one equation of a

system of two equations is of the first degree, the solution can

be obtained by the method of substitution.

Ex. Solve the system y + 2 x = = 5,

(1) (2)

Substituting 5-2 x for y in (2),

x2-25+20x-4x2-8.

It is proved in School Algebra, Ch. XXIV., that the above method is based upon equivalent equations.

Therefore the solutions of the given system are 1, 3; 5, -63, the first number of each pair being the value of x, and the second the corresponding value of y.

Had we substituted 1 for x in (2), we should have obtained y= ± 3.

But the solution 1,3 does not satisfy equation (1).

Therefore, always substitute in the linear equation the value of the unknown number obtained by elimination.

3. Elimination by Addition and Subtraction. - This method can frequently be applied.

The given system has the two solutions 3, 3; -3, 3.

Notice that this example could also have been solved by the method of substitution.

tain the unknown numbers in both equations of the system are of the second degree, a system can always be derived whose solution is obtained by the method of Art. 2.

or

or

Ex. Solve the system

x2 + xy + 2 y2 = 74,

(1)

2x2+2xy + y2 = 73.

(2)

(3)

(4)

Multiplying (1) by 73, 73 x2 + 73 xy + 146 y2 = 74 × 73.

Multiplying (2) by 74, 148 x2 + 148 xy + 74 y2 = 74 × 73.

Subtracting (3) from (4), 75a2 +75 xy-72 y2 = 0,

The solutions of these systems, and hence of the given system, are respectively 3, 5; -3, -5; 8, -5; — 8, 5.

In applying this method to such systems, we must first derive from the given equations a homogeneous equation in which there is no term free from the unknown numbers.

5. Such examples can also be solved by a special device.

and from (2),

Then from (1), x2 + 4 x2t2 = 13, whence a2

x2t+2x22 = 5, whence x2

9, whence x = ± 3.

When t, 22, whence x = ±√1⁄2.

=

When x = : ± 3, y = tx = } ( ± 3)= ± 1.

When x, y = − { (±√/}) == {{√}•

=

6. Symmetrical Equations. A Symmetrical Equation is one which remains the same when the unknown numbers are interchanged.

A system of two symmetrical equations can be solved by first finding the values of x + y and x y.