3d. SYMMETRICAL EQUATIONS. When the equations are of this description, they may be solved by taking advantage of multiple forms, and of the necessary relations existing between the sum, difference, and product of two quantities. 3. Given x+y = 10 to find the values of x and y. wy: OPERATION. XY = 21. x+y = 10, (1) (2) Squaring (1), +2xy +y = 100; subtracting 4 times (2), x'2xy +y* = 16; by evolution, 2-y = 4; but in (1), x+y = 10; ; whence, x = 7 or 3; 3 or 7. x 4. Given to find the values of x and y. 3 + xy +y = 133 } OPERATION. (1) (3) p=6; Put XY =p. s+p= 19, s'-p=133. Dividing (2) by (1), 8-p=7; whence from (1) and (3), s= 13, and x+y=13, and xy = 36. Proceeding now as in the third example, we have 2+2xy +y = 169, x-y=+5, x=9 or 4, y=4 or 9 that is, 2 Put EP, and y}=Q; then P’, and y 23 Q. Substituting these values in the given equations, P+Q=6, (1) P: +Q=20. (2) From (1) P: +2PQ+Q=36; (8) taking (2) from (3), 2PQ = 16; (4) P-Q=+2; P=4 or 2; Q 2 or 4. whence we have yš= 2 or 4; x = 64 or 8, y = 32 or 1024. 2 = +8 or +272. In this example, the auxiliary letters were used to avoid fractional exponents in the operation. This practice, however, is not a necessity, but only a convenience. The auxiliary letters should be made to represent the lowest powers of the unknown quantities. 4 or 2, P, Assume 20 y=Q; then, y=Q"; and 2 * = P', yo Substituting these values in the given equations, and factoring, P'(P+Q= 208 = 13-16, (1) l'(Q+P)=1053 = 13.81. (2) P't Q" 81 (8) 9 ; 9P 4Q or, Q and P= 9 Substituting these values in (1) and (2), we have 9PS (0) 4Q ♡ P' ==64; Q=y=729; whence, : +8, and If we take the minus sign in the second member of equation (4), we shall obtain +8V-4:y= +271—. x+y= 8 7. Given to find the values of x and y. ** +yo 152 from (7), y=+27. OPERATION. XY = 15. x+y = 8, (1) x+y = 152. Cubing (1), ** +3x*y+3xy+y = 512; (3) taking (2) from (3), 3x*y+3xy' = 360, xy(x+y) = 120; (5) dividing (5) by (1), (6) Whence, by combining (1) and (6) as in the 3d example, x = 5 or 3, y 3 or 5. 300. For examples of more than two unknown quantities, no additional illustrations are necessary. The few cases which lead to a final equation in the quadratic form are to be treated by the same methods that apply to the preceding. And skill in the management of this whole class of examples, must depend less upon precept than upon practice. 301. As ixiliary to the solution of certain questions, particularly in geometrical progression, we give the following PROBLEM.-Given x+y: =s and xy = p, to find the values of xo +y?, x+y', '+y', and x'+y', expressed in terms of s and p. SOLUTION. x+y = s, (2) Squaring the first, #*+2xy +y = 8"; 2xy = = 2p; 1st result, xo+y= $_2p. (A) Multiplying (A) by (1), 2+x*y+xy'+y = 3-2ps; subtracting xy(x+y) ps; 2d result, x+y = 3-3ps; (B) Again, squaring (A), x+2x"y"+y=8-48"p+4p'; subtracting 2x"y' 2po; 3d result, 3* +y*=s_4sop+2p*. (0) Multiplying (4) by (B), +x*y*+x*y* +y' = so— 5sop+6sp"; subtracting a’y*(x+y) sp?; 4th result, 20+y* = 50-55*p+5sp". (D) The following example will illustrate the use of these formulas. +y=: 9 to find the values of x and y. X*+y* = 2417 In this example we have 9, sa - 81, s4 6561. Hence, from (C), we have 6561 - 324p+2p' = 2417; pe-162p = -2072, p-162p+6561 = 4489, p-81 = +67, xy = p = 148 or 14. If we take xy = 148, the values of x and y will be imaginary. Taking xy = 14, with the equation x+y = 9, we readily obtain x = 7 or 2, y = 2 or 7. 1. Given { S = 15 16} { 17, Ans. { { 25} Ry=+3. Ry= 25. EXAMPLES OF SIMULTANEOUS EQUATIONS. Find the values of the unknown quantities in the following groups of equations : 1. my sa = 18 or 121, Ans. x-2y = 039 y= 8 or -21. 2. 5 xy +2y = 120 x = 8 or Ans. 2.++y y=6 or -13. 3. = 25 x=9 or – 1418, 4x =9y y=4 or 61. 4. 5x*—y=35 Ans. x = 3 or -4, 15x +y=25 10 or 45. 5. $ 4x+3y = 43 Ans. x = +2, 3x*—y: = 3 6. 3.0*+xy = 336 28 or 12, Ans. 40 -72 or 8. 7. { xy +yo = 126 x = +15, Ans. 15(x+y) = 7:30 8. {2+4y* = 161) Ans. x = +9, 15(2-y) = 4y 9. $**+xy = 12 Ans. = +2, y*+xy = 24 $ ac" Ans. s x = EV, x+y = 2 271 11. x*+xy = 56 Ans. sa= +472 or +14, xy+2y = 60 y=+31/2 or 710. 3x+xy = 681 x=+4 or +34V3, Ans. 4y + 3xy = 160 y=+5 or +10v3. 8 x = +3 or + 13. Ans. 1 xy—2y = 15 y=+1 or } { } } y = + 6. Ans. 17,5 74. y = 12. 20* +xy |