2x 2cm And, if : a :: 6 :c; then will =ab, or 2cx= 3 3 3ab 3ab ; or, by division, x= 20 43 Also, if 12 4:1; then 122 2 12 or 2x+x=12 ; and consequently x= -4. 3 = 2x, X: : MISCELLANEOUS EXAMPLES. 1. Given 5x - 15=2x+6, to find the value of x. Here 52- 2x=6+15, or 3x=6+15321 ; and there 21 fore x 3 2. Given 40-62-16=120-14x, to find the value of c. Here 14x - 6x = 120-40+16'; or 8x = 136-40 96 96 ; and therefore x=3 =12 3. Given 3x2 - 10x=8x+x", to find the value of x. Here 3.x—10=8+r, by dividing by w; or 3x -x=8 +-10=18, by transposition. 18 And consequently 2x=18, or x== =9. 2 4. Given 6ax3 12abr2 -3ax3 +6axa, to find the value of x. Here 2x - 46=2+2, by dividing by 3axo; or 2x-X =27-4b; and therefore c=46+2. 5. Given x2+2x+1=16, to find the value of Here +1=4, by extracting the square root of each side. And therefore, by transposition, x=4-1=3. Here 5ax-2dx=c+3b; or (5a - 2d)x=c+36 ; and therefore, by division, a = c+36 5a-2d" 2 2 2 3 4 6x = 60; or 4 12x - 8x+6x=240 ; whence 10x=240, or x=24. + - 19 2 3 2 8. Given + 2 3 20 to find the value of x. 2x Here 2-3+ =40-2+19; or 3x - 9+2x=120 3 3x+57 ; whence 3x + 2x+3x=120 + 57+9; that is 8.3=186, or x=231. 2x 9. Giyen V5+5=7, to find the value of %. Here v=7-5=2 ; whence, by squaring, 54, and 2x=12, or x=6. 2x 3 3 2a2 10. Given : + v(az +xo) to find the ✓(az +x2) value of x. Here</a3+x2)+a? + x3 = 2a? ; or xv(as +x2)= a? – ,and x2(a2 +*2)=04 - 2ax3 +x4; whence ao 23 + x4=24 - 2a2x3 + x4, anda? x2 =a4- 202x; therefore3a222 a 4 a? =a4, or x2= ; and consequently Ver 3a2 3 1 3 Fan V3, the answer required. 3 a 3 9 2 1. Given 3x – 2+24=31, to find the value of x. Ans. x=3 2. Given 4-9y=14-1ly, to find the value of Y. Ans. y=5 3. Given x+18=3x - 5, to find the value of x. Ans. x=113 4. Given x++-=11, to determine the value of x. 3= Ans. x=6 23 +1=5x – 2, to find the value of x. 10 5. Given 2x 6 Ans. x= 3 7 6. Given + to determine the value of x. 2 3 4 1 Ans. x=1 5. x+3 -5 to find the value of x. 6 Ans. x=3 13 8. Given 2+73c=74-+-Ex, to find the value of x. Ans. x=12 x2 9. Given uta= to find the value of x. uta Ans. x=2a 2 to find the value a 10. Given vätvats=vlatz) of ton, ax b a bx 11. Given + bxa 4 to find the value 36 Ans. x= 30-26 of * 12. Given Va+*=Vb4+x", to find the value of x. 64 -0% Ans. 2= 2023 of x. 13. Given va+x+va-=vax, to find the value 402 a2 +4 Ans. of x. 15. Given ata=vaz.txx(62+x2), to find the value 62 Ans. 4a 16. Given iv x2 +3a%-ivX3 --3a2=Xva, to find the 9a3 value of x. Ans. x=> 4 4 17. Given vatx+va-a=b, to find the value of x. 6 Ans. 2= 18. Given vata+va-=b, to find the value of x Vas 3 Ans. x = 63 - 2a 36 a 19. Given vat vu=vax, to find the value of x. Ans. x = (va-1); :+1 1 20. Given + =a, to determine the of x.. a 21. Given as tax=a-vaz-ax, to find the value Ans. x= V3. 22. Given vaz - x tavas -1=a? V1 -22, to find the value of x. Ans. I= a2 +3 Given vata=c-vx+b, to find the value of x. cat-bAns. x= ( -6 6 4bc 24. Given ✓ tváre ata a? the value of x. Ans. X=a to find Of the resolution of simple equations, containing two unknown quantities. When there are two unknown quantities, and two independent simple equations involving them, they may be reduced to one, by any of the three following rules : RULE I. Observe which of the unknown quantities is the least involved, and find its value in each of the equations, by the methods already explained ; then let the two values, thus found, be put equal to each other, and there will arise a new equation with only one unknown quantity in it, the value of wbich may be found as before. (6) (6) This rule depends upon the well known axiom, that things which are equal to the same thing, are equal to each other; and the two following methods are founded on principles which are equally simple and obvious, |