31. At what time between 8 and 9 o'clock are the hands of a watch 5 minutes apart?

32. Find the H. C. F. and the L. C. M. of

x2-4, 8—23, −4 + 4 x − x2, (2 − x)2, and (x − 2)(1 -- x).

4x2 m2 4 2x+m

20+1

34. Simplify (1+1+1)+~+1+(-1)+1)

23-1

35. What value of x will make (4x+3)(3x-1) equal to (6x+5)(2x-1)?

36. Simplify

(b-c)(c-a)(a-bb-c
+
(b+c) (c+a) (a + b) b + c

+

c-a

+ c+ a a+b

37. By what must +22+3x2+4x+5 be divided to give a quotient of x2+4x+14 and a remainder of 44x+47 ?

43. The head of a certain fish weighs h pounds, the tail weighs as much as the head and the body, and the body as much as the head and tail. What is the weight of the fish

46. Solve for m: (2 m + 2) (3 b −c) +2 ac=2 c(a — m).

47. If a certain number of wagons is sold at $80 each, the same amount is received as when 10 less are sold at $100 each. How many are sold in each case?

49. Factor x2 + a2 — 2(1 — ax) — (x + α).

m+1.

m

50. Two bills were paid with a 10-dollar bank note. One bill was 25% more than the other, and the change received was the smaller bill. Find the amount of each bill.

51. Solve for s:

(s − a) (s + b) — (s + a) (s — b) — 2 (a — b) = 0

52. Factor 26-203 — 2 (3 x3 + 4).

53. Solve for n, l = (n − 1) (a − 1) + a.

54. An orderly is dispatched with an order, and 3 hours after he leaves a second orderly is sent after him with instructions to overtake the first in 6 hours. To do this he must travel 4 miles an hour faster than the first traveled. How many miles an hour does each travel?

CHAPTER XVI

SIMULTANEOUS LINEAR EQUATIONS. PROBLEMS

183. If x and y are two unknown quantities and their sum equals 7, we may write

x+y=7.

Clearly, an unlimited number of values of x and y will satisfy this equation. For example:

184. Such an equation in two unknown quantities, satisfied by an unlimited number of values for the unknown quantities, is an indeterminate equation.

185. If, however, we have with this equation a second equa tion stating a different relation between x and y, as

is such that each is satisfied only when x = 5 and y = 2.

No other values of x and y will satisfy this pair of equations. Hence,

186. Simultaneous equations are equations in which the same unknown quantity has the same value.

187. A group of two or more simultaneous equations is a system of equations.

188. Two possible cases of equations that the beginner may confuse with simultaneous equations must be carefully noted. (a) Inconsistent Equations.

Given: x + y = 9,

x + y = 2.

It is manifestly impossible to find a set of values for x and y that shall satisfy both given equations. The equations are inconsistent.

If the first equation is multiplied by 3, it becomes the same as the second equation, and every set of values that satisfies the second satisfies the first as well. The equations are equivalent.

189. For a definite solution of a pair of simultaneous equations we must have a different relation between the unknown quantities expressed by the given equations.

190. Equations that express different relations are independent equations.

191. Simultaneous equations are solved by obtaining from the given equations a single equation with but one unknown quantity. This process is elimination. Each of the three methods of elimination in common use should be thoroughly mastered.

From the illustration we have the general process for elimination by substitution:

192. From one of the given equations obtain a value for one of the unknown quantities in terms of the other unknown quantity. Substitute this value in the other equation and solve.

The method of substitution is of decided advantage in the solution of those systems in which the coefficients of one equation are small numbers. In later algebra a knowledge of this method is indispensable.

In applying this process of elimination care should be taken that the expression for substitution is obtained from the equation whose coefficients are smallest. The resulting derived equation will usually be free from large numbers.