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OF QUADRATIC EQUATIONS,
WITH TWO UNKNOWN QUANTITIES.
176.) For the solution of Quadratic Equations with two unknown quantities, the three following methods, which are generally used, will be found applicable to a vast variety of questions.
The FIRST METHOD is performed by this general
(77.) Rule.- Find the value of one unknown quantity in terms of the other, in one of the given equations, and substitute this value for it in the other equation ; from which there will arise a quadratic equation soluble by the methods already given.
Eramples. 2.r+y '1. Given
to find the values of x and y. and 3ry = 210 Solu.-Multiply by 3, and from the first equation 2x +y=27.
27 .. 2r=27-y; and rz
Or, 3 (27-y) xy=420, multiplying by 2.
Or, .. 27y-yo=140, dividing by 3
729_ 729 729-560 Comp. the sq. ye_274 + 4
= 20 or 7;
34 or 10. 2
2 2. A certain number consists of two digits ; the left hand digit is equal to three times the right hand digit; and if we subtract 12 from the number itself, the remainder will be equal to the square of the left hand digit; it is required to find the number. Solu.-Put I =left hand digit then will 10x +y be
and y=right band one the number.
Hence, by the question, r=3y,
and 10.r + y-12=r; Therefore, 30y +y-12=9ys by substitution; Or, gyo-31y=-12;
12 Whence yo -y=
-= 3 or
3. Let there be two numbers such that, if the less be taken from 3 times the greater, the remainder shall be 35 ; and if 4 tines the greater be divided by 3 times the less plus 1, the quotient will be equal to the less number; to find these two numbers. Ans. 13 and 4.
4. There is a certain number, and the sum of its digits is 15; now if 31 be added to their product, the digits will be inverted ;. it is required to find that number.
and y =
5. Given ty= 8
to find the values of I and y.
Ans. x = 16,
(78.) The SECOND METHOD :
When those parts of the given equations that contain the two unknown quantities bear a particular relation to one another; as, for instance, their sum and difference; their sum and product; their sum and the sum of their squares; their product and the sum of their squares; their sum and the sum of their cubes, &c. &c. The equations may then be managed by some one of the following solutions.
Then, the square of the first equation, =r? + 2.ry+y'=a' (A) And the second equation x by 4
=46 (B). By subtracting (B) from (A) .—2xy +ye=a2-40. Extracting now the square root 1-y=+va-41,
x+y=a. Hence, by addition, 2x=a+va-4b, or rs
a F var-46 And, by subtraction, 2y=a F va-46, or y= (81.) 3. Let x +y =a
Then the square of the first equation is ze + 2xy +yo=a* (A); From which take
2 +y==; Therefore
2xy=a'-6 But multiply by 2, and then we have 4xy=2a'—26 (B). Subtract (B) from (A), and x-2xy + y'=a-2a-26=26-a.
Hence, 1-y=+_26—a. Having now x+y and s-y, we should proceed by the first Example fArt. 79) to find the values of x and y. (82.) 4. Let **+y=a
Then will x+3xy +3.xy? +y*=a , cubing the first equa
+y=1, the second equation
Then the difference will be 3.roy + 3xy'=a'-6,
Having now found x+y and xy, the values of x and y may he found as in the second Example.
Raise rty to the 4th power, and x* +4x*y+6roy? + 4xyo+y=a;
Let now atub=0
Divide the remainder by xy, and we have 4x? + Oxy +4y' =
xy Now the first equation squared is .... ...+ 2xy + y=a; Therefore,
4.x2 +8xy +4yo=4a* (B) From (B) the greater take (A) the less, then 2xy=4a?
Hence we have ry—a®= Va-c,
X+5x*y +10.x*y* +10x®y + 5zy'+y=a';
5xy x 2 + 2x®y + 2xy! + ys=c;
And then x and y may be found as in Example 2. (86.) 8. Given x+y=s, and xy=p; to find the value of x +yo, 1+y'; **+y* ; &c. in terms of (s) and (p).
First, the square of the 1st equation, x + 2xy +yo=s* ;
(ro+y') x (x+y) = (5—2p),
x+y = 5-3ps. And, thirdly,
(r+yo) x (x+y) (5-3ps), Or, which is the same, **+ry x (+y)+y = 5*—3pse,
That is, x+px (so—2p)+y* St-3ps,
r* +y = $-4ps'+2ps.
Examples for Practice. 9. There are two numbers, the sum of which is 19, and the surr of their squares is 193 ; required those numbers.
Ans. 12 and 7 10. The sum of the squares of two numbers is 261, and their pro'duct is 90; required those two numbers.
Ans, 15 and 6 11. The sum of two numbers is 9, and the sum of their cubes is 351 ; required those two numbers.
Ans. 7 and 2 12. The sum of two numbers is 7, and the sum of their 4th powers is 641; what are those numbers ?
Ans, 5 and 2 13. The sum of two numbers is 6, and the sum of their 5tt powers is 1056; what are those numbers ?
Ans. 2 and 4. 14. The sum of two numbers is 13, and their product is 30; it is required to find the sum of their 4th powers.
Ans. 10081. à (87.) The THIRD METHOD.- Equations of the form
Odry + eys in which the sum of the indices of the unknown quantities is the same in every term of both equations, are managed by