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Ex. 12. Giveo 2x+3y=118, and 5x2 – 7y=4333, to find the values of x and y.
3268 Ans. =35, or
17 Ex. 13. Given 2? +4y*=256 – 4xy, and 3y - 2*=39, to find the values of x and y.
Ans. x=+6, or + 102 ; and y=+5, or +59. Ex. 14. Given 2" tyń=2a", and xy=c>, to find the values
of x and y.
x=[a Ivica -c)];
*"á : Aps.
[a" Evla" - c)] Ex. 15. Given r?+2xy + y +22=120—2y, and xy - y=8, to find the values of x and y.
Ans. x=6, or 9, or-97V5; and y=4, or 1, or-3+
S and y
Ex. 16. Given x2+ ye_2—y=78, and xy+x+y=39, to find the values of x and y.
Ans. x-9, or 3; or-61+;V-39 ; and y=3, or 9, or61 Fi-39.
4x 85, Ex. 17. Given +
to find the values of x and y. y and -y=2, 17
3 Aos. =5, or
; and y=3, or 10
10 Ex. 18. Given x-2xy +y=49, to find the values of a and 24 - 2x^y tyø-x+y=20,
Ans. x=+3, or+/6, or + V (30+6/5);
and y=2, or—1, or ș(1+3/5).* Ex. 19. Given
3-y, and 4-x=y-y2, to find the values of x and y. Ans. x=4, or \ ; and y=1, or 21.
Ex. 20. Given wi+e-4x=y"+y+2, and xy=y+3y, to find the values of x and y.
Ans. x=4, or 1 ; and y=i, or -2. Ex. 21. Given x2 + xy=56, and xy +2y=60, to find the values of x and y.
Aps. x=+472, or F14;
and y=+372, or£ 10. Ex. 22. Given x-y=15, and xy=2yy, to find the values
Ans. x=18, or 12] ; and y=3, or-2. Ex. 23. Given 10x+y=3xy, and 9y-9x=18, to find the values of x and y. Ans. x=2, or -; and y=4, or ģ.
There are four other values, both of x and y, which are all imagi,
of x and y.
Ex. 24. Given xty: 2-y:: 13:5, 7 to find the values
and y +=25. of x and y.
Ans. x=3, or--141 ; and y=4, or-61. Ex. 25. Given toy-7xyo=1710, and xy-y=12, to find the values of and y.
-19 Ans. x=5, or š, or
; and y=3, or -15, oro
, 17+61 - 2 6+7-2.
Ex. 26. Given xy t-xy?=12, and story:==18, to find the values of r and y.
Ans. x=2, or 16 ; and y=2, or Ex. 27. Given x+y+t+y)=6, and 2+3=10, to find the values of x and
y. Ans. r=3, or i'; or 44+1-61; and y=1, or 3, o 4tiv81.
Ex. 28. Given 2? +4V(x+3y+5)=553y, and 6x-74 =16, to find the values of x and y.
-70+/3895 y=2, or
49 Ex. 29. Given 2? +-2xy=441—x'y, and xy=3+x, to find the values of x and y.
I=3, or-7,; or-2+/-17,
y=2, or 4 ; or -17. Ex. 30. Given (x+y)2—3y=28+32, and 2xy +3x=35, to find the values of x and
Sr=5. or, or-s+v(-255)
y=2, or ;, or-FV(-255.) Ex. 31. Given 32+3x+y=73-2xy, and y+3y+z=44, to find the values of x and y.
x=4, or 16 ; or-127 58,
y=5, or-7 ; or-1/58. Ex. 32. Given =1361-2xy, and x+y=10, to find the values of x and
x=6, or 4; or 5+57(-i).
y=4, or 6; or 575/11). Ex. 33. Given ye_432=12ry, and yo=1242xy, to find the values of x and y.
Ans. x=2, or 3 ; and y=6, or ✓(21)+3.
THE SOLUTION OF PROBLEMS,
PRODUCING QUADRATIC EQUATIONS.
§ I. SOLUTION OF PROBLEMS PRODUCING QUADRATIC EQUATIONS,
INVOLVING ONLY ONE UNKNOWN QUANTITY.
428. It may be observed, that, in the solution of problems which involve quadratic equations, we sometimes deduce, from the algebraical process, answers which do not correspond with the conditions. The reason seems to be, that the algebraical expression is more general than the common language, and the equation, which is a proper representation of the conditions, will express other conditions, and answer other suppositions.
Prob. 1. A person bought a certain number of oxen for 80 guineas, and if he had bought four more for the same sum, they would have cost a guinea a piece less ; required the number of oxen and price of cach.
80 Let := the number ; then = the price of each ;
80 x +4
- 1, by the problem,
•'.x=16, or -20.
=5 guineas, the price of each.
16 The negative value (-20) of x, will not answer the condition of the problem.
Prob. 2. There are two numbers whose difference is 9, and their gum multiplied by the greater produces 266. What are those pumbers ? Let x= the greater; .. 8-y= the less.
9 266 and x.(2x-9)=266 ; ..x
9 47 completing the
4 ..x=14, or -91 ; and a
9=5, or -18) Here both values answer the conditions of tbe problem.
Prob. 3. A set out from C towards D, and travelled 7 miles a day. • After he had gone 32 miles, B set out from D towards C, and went every
day one-nineteenth of the whole journey; and after he had travelled as many days as he went miles in one day, he met A. Required the distance of the places C and D. Suppose the distance was a miles.
= the number of miles B travelled per day; and also 19 = the number of days he travelled before be met A.
361 by transposition and completing the square,
+36=36-32=4; 361 19
-8, or 4 ; and x=152, or 76, both which values an
19 swer the conditions of the problem. The distance therefore of C from D was 152, or 75 miles.
Prob. 4. To divide the pumber 30 into two such parts, that their product may be equal to eight times their difference.
Let x= the lesser part ; 30—-= the greater part, and 30-X-X, or 30~2x= their difference.
Hence, by the problem, x(30-2)=8(30—2x), or 301-22 -240-16x; ..2---46x=
-210. Completing the square, x_46x+529=289;
...X=23+17=40, or 6= lesser part ;
and 30—3=30—6=24= greater part. In this case, the solution of the equation gives 40 and 6 for the lesser part. Now as 40 cannot possibly be a part of 30, we take 6 for the lesser part, wbich gives 24 for the greater part ; and the two numbers, 24 and 6, answer the condi. tions required.
Prob. 5. Some bees had alighted upon a tree; at one flight the square root of half of them went away; at another eight-ninths of them; two bees theu remained. How many then alighted on the tree?
or 9.x+16.2°+18=18x 1.2.1-9x=18; (Art. 417), Multiplying by 8, 16x*—72x=144 ; adding 81 to both sides, 16x9_72x+81=225; :::4x=9+15=24, or --6; and x=6, or -11.
.::2x2=72, or 4. But the negative value -1 of x, is excluded by the nature of the problem ; therefore, 72= number of bees.
429. If, in a problem proposed to be solved, there are two quantities sought, whose sum, or difference, is equal to a given quantity, for instance, 2a ; let half their difference, or half their sum, be denoted by x; then xta'will represent the greater, and xva the lesser, (Art. 102). According to this method of notation, the calculation will be greatly abridg: ed, and the solution of the problem will often be rendered very simple.
Prob. 6. The sum of two numbers is 6, and the sum of their 4th powers is 272. What are the numbers ?
Let x= half the difference of the two numbers ; then 3+ x= the greater number, and 3-x= the lesser.
.. by the problem, (3+x)+(3--x)=272, or 162+108x2+2x9=272; from which, by transposition and division, 24 +5422=55 :
... completing the square, x + 54.xo+729=784,
and extracting the root, x? +-27=+28;
...x=-27+ 28, and x=+1, or IV-55. Now, by taking the positive value, +1, for x, (eince in this case, it is the only value of x which will answer the problem); we shall have 3+1=4= the greater, and 3-1=2= the lessér.
Prob. 7. To divide the number 56 into two such parts, that their product shall be 640.
Ans. 40, and 16, Prob. 8. There are two numbers whose difference is 7, and half their product plus 30, is equal to the square of the lesser number. What are the numbers ?
Ans. 12, and 19. Prob. 9. A and B set out at the same time to a place at the distance of 150 miles. A travelled 3 miles an hour faster than B, and arrives at his journey's end 8 hours and 20 minutes before him. At what rate did each person travel per hour ?
Ans. A 9, and B 6 miles an hour ? Prob. 10. Tbe difference of two cumbers is 6; and if 47 be added to twice ihe square of the lesser, it will be equal to the square of the greater. What are the numbers ?
Ans. 17, and 11.