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XX. 1. mx+y-a, x-my= b. 2. ax_by = a, bxay = b?. 3. (a - b)x+(c-d)y=p+q, (a+b)x-(c+d)y=p-9. 4. ax by = m, cx+dy = n. 5. ax+by =c, px = qy. 6. a(x - y)+bx+y)+cx-cy=d, x-y-1. 7. (a+b)x+(6+c)y=d, (5-0):-(c-by = e. 8. ax+by = xy = bxay. 9. axy = c(bx +ay), bxy = c(ax-by). y

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XXI. 1. Find two numbers such that the sum of twice the first and the Bocond is equal to 17, and the sum of three times the second and the first is equal to 19.

2. Find two numbers such that half of the first and three-fourths

of the second together are equal to the difference of three times the first and the second, and this difference is equal to 11.

3. Two digits, which form a number, change places on the addition of 9, and the sum of these two numbers is 33; find the number.

4. There are two numbers, each consisting of three figures. If unity be added to the sum of the two numbers the amount is 1,000. If the smaller number be written as a decimal fraction after the greater, the result is six times the number composed of the greater number written as a decimal fraction after the smaller; find the two numbers.

5. If the less of two numbers be divided by the greater, the quotient is 21 and the remainder is .04162. If the greater be divided by the less the quotient is 4 and the remainder :742; find the numbers.

6. Find two numbers, such that if s of the less be added to of the greater, the sum will be 7; but if } of the greater be taken from the less, the remainder will be 2.

7. Divide the number 208 into two parts such that the sum of onequarter of the greater and one-third of the less part when increased by 4 shall be equal to four times the difference of the parts.

8. There is a fraction such that if 5 be added to its numerator the value of the fraction is doubled, and if 4 be taken from its denominator the value of the fraction is trebled; find the fraction.

9. Find a fraction such that when its numerator is diminished by unity, two-thirds of it is equal to 9; and when the denominator is diminished by unity, one-seventh of this shall be equal to 2.

10. There is a fraction such that when its denominator is increased by one the fraction becomes one, but when the numerator is increased by 1 the fraction becomes }; find the fraction.

11. What fraction is that whose numerator being doubled and the denominator increased by 7, the value of the fraction becomes ž, but if the denominator be doubled and the numerator increased by 2, the value becomes 2.

12. Solve the equations aix+by=C1, 2,2+buy = Cz. What form do the values of x and y take when 94-1, 4, and what does it indicate ?

a. 6, C2

XXII. 1. A sum of money consists of half-crowns and sixpences, and is. worth as many shillings as there are pieces of money, also the square of the number of sixpences is 36 times the number of half-crowns; find the sum of money.

2. On a division of the House of Commons, if thə number of members for the motion had been increased by 40 from the other side, the question would have been carried by 3 to 2; but if those against the motion had received sixty of the other party, the motion would

have been lost by 2 to 1. Did the motion succeed, and how many members voted on the question ?

3. A certain number of men do a piece of work in a certain number of days; had there been 4 men more they would have finished the work in 8 days, but three times the number of men would have finished the work in two days less than half the time employed; required the number of men actually employed.

4. Show how £12 can be paid by means of guineas and moidores, so that the total number of coins which pass in the transaction shall be 16.

5. A's annual income is half of B's, and B spends £60 a year more than A. At the end of two years A has saved $200, and B £600. What are their yearly incomes ?

6. A pound of tea and three pounds of sugar cost six shillings; but if sugar were to rise 50 per cent., and tea 10 per cent., they would cost seven shillings. What were the prices of tea and sugar?

7. Two plugs are opened in the bottom of a cistern containing 192 gallons of water; after three hours one of them becomes stopped, and the cistern is emptied by the other in eleven hours; had six hours occurred before the stoppage it would only have required six hours more to empty it. How many gallons will each plug-hole discharge in an hour, supposing the discharge uniform ?

XXIII.

1. x+y = 5, +% = 4, y+%= 8.
2. x+3—%= 8, x+%-y=9, y+*-* = 10.
3. 2.: +3y = 13, 5x+4z = 26, 4y+3z = 24.
4. x+100 = y +z, y+100 = 2x + 2z, z+100 = 3x+3y.
5. x-y--= 6, 3y--—% = 12, 72-y-2 = 24.
6. x+y-2 = 8x+3y-6z = 32 - 4x-y=1.
7. 5(2x+3y) = 11(x+y), 3(x+3) = 5(2-2), 14(104—3z)

= 37(4y-2). 8. 3x+2y—% = 20, 2x+3y+6= 70, x-Y+63 = 41. 9. x+2y + 3z = 17, 2+2x+3y = 12, y+23 + 3x = 13. 10. 2x+3*y+4z = 70, 5'x+6*y+ 722 = 224, 8*x+9y+10*2 = 526. 11. 2x+3y-82+35 = 0, 73-4y+5-8 = 0, 12x-5y—3z +10 =P 12. y y

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XXIV. 1. ax+y=c, by +2 = a, cz+x= b. 2. c(x+y) = az, cx+2) = by, x+y+: = c.

3. a(x+ = bz, by+c) = az, bextacy+ab% = 0. 4. ay+bx = 0, bz+cy = a, cx+az = b. 5. x+y+z = a, bx+cy+dz = 0, b*x+y+dʻz=0. 6. 2+3 = 4, (@+5)x+(b+c)3 =(d+c), abc+bc3 =1+acy. 7. x+y+z=0, (a+b)x+(a +c)y+(6+c)2 = 0, abx +acy + bcz = 1.. 8. ax+by+cz = a*, bx+cy + ax=b?, cx+ay+bz = c. 9. (b+c)x+(a+c)y+(a+b)2 = 9, x+y+s=p, box+acy +abz = r. 10. ax+by+cz = P, a*x+by+c^2 = 9, aRx+by+02= r. 11. ax+(b+c)y+(6c)z = 6, bx+(c+a)y+(c—a)z = 0,

cx+(a+by+(ab)x= a. 12. ax+cy + b2 = cx +by+az = bx+ay+cz = a +b+-3abc. 13. xy = a(x+y), xz = 5(x+z), z = c(x+y). b 6

6 14.

++ = 1. y y

15. xy = m(ay+bx), yz = n(cz+dy), xz=p(ez+fr). 16. ax+cy+b3=a? +2bc, cx+by+az=b: +2ca, bx+ay+02 = c +2ab.

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XXV. Eliminate x, y, z from the following equations :1. x = a(y+z), y = 5(x+x), %=c(x+y).

2. (b+c).x+(c+ay+(a+b)x= 0, (5-0)x+(-a)y+(a-b) = 0, and 2-1+y!+--1-0.

3. ax+by+cz = bx+cy +az = cx+ay+bz= 1, when x+y+=*= p. 4. mn(2-a) - In(y-B) - Im :-)); m'n' (x'—x) - In(y-y)

=lm' '-). 5. bx+cy+az=ax+ay+b:= ab+bc+ac, x+y+z= a+b+c.

XXVI. 1. Determine three numbers such that their sum shail be 9, the gum of the first, twice the second, and three times the third shall be 22, and the sum of the first, four times the second, and nine times the third shall be 58.

2. The differences between the first and second, and the second and third digits of a certain number are equal. If the number be divided by the sum of the digits in the units and tens place, the quotient is 107; but if 396 be subtracted from the number its digits will be inverted. What is the number?

3. A certain number of three digits exceeds the sum of the digits by 180. If the digits be reversed it exceeds the sum by 378; but if divided by the sum of the digits, the quotient is 14 and the remainder is 11. Find the number.

4. A number consisting of three digits is equal to 42 times the sum of the middle and left-hand digits, the sum of the digits is equal to 9, and the right-hand digit is twice the sum of the other two. What is the number?

5. The sum of three numbers is (p+1)(2+1)n, the sum of the two larger is equal to p times the least, and the sum of the two smaller is 9 times the greatest. Find the numbers.

6. Divide 27 into four parts such that the first increased by 2, the. second diminished by 2, the third multiplied by 2, and the fourth divided by 2, shall give equal results.

7. To find four numbers such that the sum of the first, second, and third shall be 13; the sum of the first, second, and fourth, 15; the sum of the first, third, and fourth, 18; and, lastly, the sum of the second, third, and fourth, 20.

XXVII. 1. A and B together possess £120, B and C £150, and C and A £140. What does each possess ?

2. A person has 200 coins consisting of guineas, half-sovereigns, and half-crowns; the sums of money in guineas, half-sovereigns, and half-crowns are as 14, 8, 3. Find the numbers of the different coins.

3. Handel's last oratorio, “Jephtha,” was composed in the eighteenth century. The first and last digits in the number expressing its date are the same, and if the digits be inverted the number will be diminished by 180. Find the date.

4. A sum of money which consists of shillings, half-crowns, and sixpences, is worth as many shillings as there are pieces of money; it is also worth as many half-crowns as there are shillings; and the number of sixpences is one more than the number of shillings. Find the number of each of the coins.

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