16. If a bushel of wheat occupies 14 cu. ft., find a formula for a number of bushels of wheat (W) in a rectangular bin, l feet long, w feet wide, and h feet high. Then solve this formula for h.

17. A rectangular bin is to contain 1000 bushels of wheat. If the bin is to be 24 ft. long and 5 ft. wide, find its depth by use of the formula obtained last in Ex. 16.

18. A number exceeds the sum of its third and fourth parts by 45. Find the number.

19. Copy the following and fill in the vacant place :

20. Solve a = p(1 + rt) for r and state the result as a rule. 21. Solve a = p(1 + rt) for t and state the result as a rule. 22. Any number multiplied by the reciprocal of itself gives what? Illustrate by the multiplication of some monomial by its reciprocal.

23. Write an algebraic expression which must represent an even number.

24. Write an expression which must represent an odd number.

- b.

What then are two factors of

25. Divide a3 b3 by a a3 - b3? From these results form a rule for factoring the difference of two cubes. By use of your rule factor x3 — 8 y3.

26. In like manner by dividing a3 + b3 by a+b obtain a rule for factoring the sum of two cubes.

=

27. In 5.27 x — 2.16 3.72 + 8 x find the value of x to the nearest thousandth.

28. The average profit in a certain business for three years was $2400. What must the profit for a fourth year be in order that the average for the four years shall be $2500?

30. Explain the difference between the Centigrade and Fahrenheit scales in measuring temperatures. If C is the Centigrade temperature equivalent to a Fahrenheit temperature F, then C = (F-32). By use of this formula find the value of

C when F = 212°.

31. If iron melts at 2700° F., what is the Centigrade temperature at which it melts?

32. Solve C (F — 32) for F. To what use can this result be put ?

33. If tin melts at 228° C., find the Fahrenheit temperature at which tin melts.

From the graph determine approximately the number of miles of railroad in the United States in the year 1875. In 1895.

CHAPTER XI

SIMULTANEOUS EQUATIONS

PART I

98. Simultaneous Equations are a set or system of equations in which more than one unknown quantity is used, and the same symbol stands for the same unknown number.

Thus, in the group of three simultaneous equations,

x stands for the same unknown number in all of the three equations, y for another unknown number, and z for still another.

99. Independent Equations are those which cannot be derived one from the other.

Thus, x+y

= 10

1. are not independent equations, since 2x=20-2y by transposing 2y in the second equation and dividing it by 2, we may convert the second equation into the first.

But 3x-2y=5] are independent equations, since neither one

5x+y=6 of them can be converted into the other.

100. Elimination is the process of combining two equations containing two unknown quantities so as to form a single equation with only one unknown quantity. Or, in general, elimination is the process of combining several simultaneous equations so as to form equations one less in number and containing one less unknown quantity.

There are two principal methods of elimination: I, addition and subtraction; II, substitution.

These methods are presented to best advantage in connection with illustrative examples.

101. I. Elimination by Addition and Subtraction.

Ex. Solve

12x+5y=75. (1)

9x-4y=33.

(2)

In order to make the coefficients of y in the two equations alike, we multiply equation (1) by 4, and (2) by 5,

Substitute for x its value 5, in equation (1),

x=5. Root.

60+5y=75,

.. y=3. Root.

CHECK.

12x+5y=12 x 5+5 x 375.

9x-4y= 9x5-4x3=33.

Since y was eliminated by adding equations (3) and (4), the above process is called elimination by addition. The same example might have been solved by the method of subtraction.

Thus, multiply equation (1) by 3, and (2) by 4,

It is important to select, in every case, the smallest multipliers that will cause one of the unknown quantities to have the same coefficient in both equations.

Thus, in the last solution given above, instead of multiplying equation (1) by 9, and (2) by 12, we divide these multipliers. by their common factor, 3, and get the smaller multipliers, 3 and 4.

Hence, in general,

Multiply the given equations by the smallest numbers that will cause one of the unknown quantities to have the same coefficient in both equations:

If the equal coefficients have the same sign, subtract the corresponding members of the two equations; if the equal coefficients have unlike signs, add.

17. If x3 and y = 4, does 12x-5y=14?

18. Make up and solve a pair of simultaneous equations

in which x = 7 and y = 2.