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For equations of this class there are in general four sets of values of x and y. It should be borne in mind that to any one of the four values of x there corresponds only one of the four values of y. Thus, when x in the 6th example is +1, y must be +3, and can not be one of the other three values given above.

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270. 3d. When the unknown quantities enter each equation symmetrically.-Substitute for the unknown quantities the sum and difference of two other quantities, or the sum and product of two other quantities.

-18 Ex. 7. Given

y

to find x and y.

2C+y=12 Let us assume

x=2+v,

y=2-v. Then

x+y=22=12 or z=6.

x=6+v and y=6-v. But from the first equation we find

2:3+y=18xy. Substituting the preceding values of x and y in this equation, and reducing, we have

432+36v2=648-18v2, whence

v=+2. Therefore x=4 or 8, and y=8 or 4.

That is,

Ex. 8. Given (20+y*=3368 )

203 +43=341

to find x and

y. Ans.

sx=3 or 5.

y=5 or 3. to find x and y.

x=5 or 6. Ans.

y=6 or 5.

Ex. 9. Given

{}

271. 4th. When the same algebraic expression is involved to different powers, it is sometimes best to regard this expression as the unknown quantity.

Ex. 10. Given

to

-=8 The first equation may be written

(c+y)+2(x+y)=120. Regarding x+y as a single quantity, we find its value to be either

10 or – 12. Proceeding now as in Art. 268, we find

a=6 or 9, or—9FV5;

y=4 or 1, or -3+75. Ex. 11. Given (4xy=96—x*y}

to find x and y.

*+y=6 Regarding xy as the unknown quantity, its value from the first equation is found to be either

8 or -12.

x=2 or 4, or 3+V21. Ans.

Ly=4 or 2, or 3 F V 21. 22 4x 85

t Ex. 12. Given 72

ya' y 9 to find x and y.

2-y=2 Regarding the

as the unknown quantity, we find its value to be either

5 17
3 3

Ans.

sæ=5 or to

ly=3 or For several of these examples there are other roots, some of which can not be obtained by the processes heretofore explained. The roots of two simultaneous equations are sometimes infinite, as in the 8th and 9th examples, where the equations may be satisfied by x=, y=FQ, since two quantities that are infinitely great may differ by a finite quantity. Solve the following groups of simultaneous equations:

sx==

Ans. x+y=2

Ty=2FV ? +

0 Ex. 14.

2

Ans. 7

6 +-=4

y

2

}

or

Ex. 13. {*°–22yyo=1}

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y=3.

x=+1V6+2a+1V6–2u. y=+1V6+2a FV6--2a.

(x=8 or 64. Ans.

y=64 or 8.

anto ty

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(+y2)xy2=

(c+y)xy= Ex. 18. xy=a

Ans. 2? +y=

x+y=72 Ex. 19.

+y=6

x?y+yéx=20 Ex. 20. 1 1 5

+
Y

4 22+y2=8 Ex. 21. 1 1 1

locatge i

205 -y=3093) Ex. 22.

2 Y=3 Ex. 23.

24 +æ?y2 +y+=931

x2 + xy + y2 =49 Ex. 24. {(7+2) (6+y)=807

x+y=5 Note. Put 7+x=2, 6+y=v.

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PROBLEMS. 1. Divide the number 100 into two such parts that the sum of their square roots may be 14.

Ans. 64 and 36. 2. Divide the number a into two such parts that the sum of their square roots may be b.

7

2° 2 3. The sum of two numbers is 8, and the sum of their fourth powers is 706. What are the numbers ? Ans. 3 and 5.

Ans: + v2a6.

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m

a

4. The sum of two numbers is 2a, and the sum of their fourth powers is 2b. What are the numbers ?

Ans. a IV-3a2+ V8a' +b. 5. The sum of two numbers is 6, and the sum of their fifth powers is 1056. What are the numbers ? Ans. 2 and 4.

6. The sum of two numbers is 2a, and the sum of their fifth powers is b. What are the numbers ?

Ans. a +

b 4a

+ -a?

10a 5 7. What two numbers are those whose product is 120; and if the greater be increased by 8 and the less by 5, the product of the two numbers thus obtained shall be 300 ?

Ans. 12 and 10, or 16 and 7.5. 8. What two numbers are those whose product is a; and if the greater be increased by b and the less by c, the product of the two numbers thus obtained sball be d?

m2 ab
Ans.
+

and
4
с
m

ab
+
2 4 C

d-a-bc where m=

с 9. Find two numbers such that their sum, their product, and the difference of their squares may be all equal to one another.

3

1 Ans.

+ and

2 4 2

that is, 2.618, and 1.618, nearly. 10. Divide the number 100 into two such parts that their product may be equal to the difference of their squares.

Ans. 38.197, and 61.803. 11. Divide the number a into two such parts that their product may be equal to the difference of their squares.

3a+avo -αφαν5 Ans.

and 2

2

,

4

12. The sum of two numbers is a, and the sum of their reciprocals is b. Required the numbers.

a
Ans.
+
4

7

a

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General Properties of Equations of the Second Degree. 272. Every equation of the second degree containing but one unknown quantity has two roots, and only two.

We have seen, Art. 250, that every equation of the second degree containing but one unknown quantity can be reduced to the form xo+px=q. We have also found, Art. 257, that this equation has two roots, viz., р

р
and

po
2
+Ve+

2

4 This equation can not have more than two roots; for, if possible, suppose it to have three roots, and represent these roots by a', a", and a!". Then, since a root of an equation is such a number as, substituted for the unknown quantity, will satisfy the equation, we must have x'2 +px'=2,

(1.) x''2+pc"=

(2.) x'"12 +px'"' =.

(3.) Subtracting (2) from (1), we have

x"22"2+p(x'- x'')=0. Dividing by x'—x", we have (ac' +'')+p=0.

(4.) In the same manner, we find (zo'+x''')+p=0.

(5.) Subtracting (5) from (4), we have

x"-"=0; that is, the third supposed root is identical with the second; hence there can not be three different roots to a quadratic equation.

273. The algebraic sum of the two roots is equal to the coefficient of the second term of the equation taken with the contrary sign.

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