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Now equation (2) gives

x? - VX=-1. Wbence x=}[v+V02–4), and x=[v— V02—4], from which, by substituting the value of v, we obtain

x=t(v5–1+/-10–2V5], and

=+-V5–1+/-10+2V5]. Hence the fifth roots of unity are

1.
7[V5-1+V-10–2V5].
7[V5–1–7–10–2V5].

-V5–1+V–10+2V5].

+(-V5-1-V-10+2V5]. Ex. 5. Find the six roots of the equation 26=1.

These are found by taking the square roots of the cube roots. Hence we have

+1,-1, 4+1V-3, -1+1V -3. Ex. 6. Find the four roots of the equation *=-1, or Q4 +1=0.

The first member may be made a complete square by adding 2x2; that is,

24 +232 +1=2x2, whence

22+1=3xV2. By transposition and completing the square,

22 ExV2+1=-1. Hence

x+1V2=+1V-2; that is,

x=1V2+IV-2,

-1V2+1V-2. These four values, together with the four values found in Ex. 3, are the eight roots of the equation

28=1.

or

EXAMPLES FOR PRACTICE.

EQUATIONS OF THE FIRST DEGREE WITH ONE UNKNOWN

QUANTITY.

7x 3x Ex. 1. Given 121 +33-6- --5%, to find the value

34 of x.

Ans. x=1394. Ex. 2. Given a (20c+195—10a)=6(x+76), to find the value

Ans. a --5a-76. 5+2 9 Ex. 3. Given 2

to find the value of a. 14

Ans. x=43.

of x.

-X

5+2=1

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X

Ans. x=

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Ex. 4. Given m+-=n-P

=p

to find the value of x. a

(n-p-m) ab Ans. x=

at 200-342-98x-27 16x-81 9 Ex. 5. Given

to find 15 20 30

24 40' the value of x.

at 64 Ex. 6. Given a3 +ab+ab2 +13= to find the value of x.

Ans. x=a-b. a2x+33

72 Ex. 7. Given

=26+; to find the value ał(ab) a?

Ans. x=a_b. 3x 3x 7x Ex. 8. Given

+

-15, to find the value of x. 5 10 4 8

of x.

Ans. x=663. 11 =

97 8

Ans. x=338 3a+2 6 Ex. 10. Given -55 to find the value of x. X

За — 6 Ans. x=

of x.

42'

m(a-) Ex. 11. Given crat

to find the value of x. 3a +

Ans. x=

am-3c+3a)

C-atm 1-2 4-5x 13 Ex. 12. Given

to find the value of x. 3 6

Ans. x=1 Ex. 13. Given (m)(n-x)=(p+x)(-9), to find the val. ue of x.

mnt pa Ans. x=

m+n+P-9

6x-22 Ex. 14. Given 8x—28=(4x+21) to find the value

3x+14 of x.

Ans. x=7. Ex. 15. Given x=a+3+

bc, cfoc

to find the value of x. à de'

Ans. x=

(ad+bc)e

de-of Ex. 16. Given -1- +3ab=0, to find the value of x.

Ans. x=

ac(1–3ab)

C- ad Ex. 17. Given (8— 3x)+(4–4x)2=(9–5x)?, to find the value of x.

Ans. x=to
16x +7
X-16

2.c +1
Ex. 18. Given
+

to find the value 24 177-98 3

Ans. x=17. 93+10 8+52 Ex. 19. Given 11-12

=1}-{x, to find the value

40 of x.

Ans. =7. 1-23c 5- 6x 8 1-322 Ex. 20. Given

to find the 3-4x 7-823.21-52x + 32227 value of x.

Ans. x=g.

a

с

20+1 ,

of x.

PROBLEMS INVOLVING EQUATIONS OF THE FIRST DEGREE WITH

ONE UNKNOWN QUANTITY. Prob. 1. Said an old miser, For 50 years I have saved 200 dollars annually; and for many years, each of my four sons bas saved annually the same sum, viz., the oldest for 27 years past, the second since 24 years, the third since 19, and the fourth since 16 years. How long since the sayings of the four sons amounted in the aggregate to as much as those of the father?

Ans. 12 years.

Prob. 2. From four towns, A, B, C, D, lying along the same road, four persons start in the stage-coach for the same place, E. The distance from A to B is 19 miles, from B to C 3 miles, and from C to D 5 miles. It subsequently appeared that the person who started from A paid as much fare as the three oth. er persons together; and the fare per mile was the same for each. It is required to determine the distance from D to E.

Ans. 7 miles. Prob. 3. Five towns, A, B, C, D, E, are situated along the same highway. The distance from A to B is 37 miles, from B to D 34, and from D to E 14 miles. A merchant at C, situated between A and D, receives at one time 8 tons of goods from A, and 6 tons from B. At another time he receives 11 tons from D, and 9 from E, and in the latter case he paid the same amount for freight as in the former, the rate of transportation being the same in both cases. It is required to compute the distance from B to C.

Ans. 15 miles. Prob. 4. If 20 quarts of water flow into a reservoir every 3 minutes, after a certain time it will still lack 40 quarts of being full. But if 52 quarts flow into it every 5 minutes during the same period, 72 quarts of water will have overflown. What is the capacity of the reservoir, and how many quarts of water must flow into it every minute in order that it may be just filled in the time before mentioned ? Ans. The capacity of the reservoir is 240 quarts, and

8 quarts must flow into it every minute. Prob. 5. A mason, by working 10 hours daily, could com. plete in a week as much over 888 cubic feet of wall as at present he completes less than 888 cubic feet, working only 81 hours daily. How many cubic feet of wall does he now com. plete weekly ?

Ans. 816 cubic feet. Prob. 6. After a certain time I have $670 to pay, and 41 months later I have $980 to pay. I settle both bills at once, at 44 per cent discount, for $1594.41. When did the first sum become due?

Ans. After 54 months. Prob. 7. A merchant gains 8 per cent. when he sells a hogshead of oil at 36 dollars. How much per cent. does he gain or lose when he sells a hogshead at 32 dollars ?

Ans. He loses 4 per cent. Prob. 8. A merchant loses 27 per cent. when he sells a bag of coffee for 39 dollars. How much per cent. does he gain or lose when he sells a bag of coffee for 411 dollars ?

Ans. He gains 38 per cent. Prob. 9. A merchant owes $2007, to be paid after 5 months, $3395 after 7 months, and $6740 after 13 months. When should the entire sum of $12,142 be paid, so that neither party may sustain any loss?

Ans. After 10 months. Prob. 10. A merchant has three sums of money to pay, viz., $1013 after 34 months, $431 four months later, and the third sum still four months later. How large is the third sum, supposing he could pay the three bills together in 67 months with- . out loss or gain?

Ans. $428. Prob. 11. A merchant has two kinds of tobacco; the one cost 40 cents per pound, the other 24 cents. He wishes to mix the two kinds together, so that he may sell it at 34 cents per pound without loss or gain. How much must be take of each sort in order to have 64 pounds of the mixture?

Ans. 40 pounds of the better sort, and 24 pounds of

the poorer.

Prob. 12. A vinegar dealer wishes to dilute bis vinegar with water. At present he sells his vinegar at 6 dollars per hogshead (120 quarts). How much water must he add to 294 hogsheads in order to be able to sell the mixture at 4 cents per quart?

Ans. 7f hogsheads.

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