3 In like manner, if ✔2x+3+4=8, then will √2x+3 =84 = 49 And 2x+3=4364, and 2x=64-3=61, or x=-3 =30 RULE 5 If that side of the equation, which contains the unknown quantity, be a complete power, it may be reduced by extracting the root of the said power from both sides of the equation. Thus, if x+6x+9=25, then will x+3=√25=5, or x=53=2. And, if 3x-9=21+3, then will 3x21+3+9= 33, and x=311, or x=√11. 2x2 In like manner, if +10=20, then will 2x2+30 3 60, and x2+1530, or *30-15=15, or =√15. RULE 6. Any analogy, or proportion, may be converted into an equation, by making the product of the two mean terms equal to that of the two extremes. Thus, if 3x 16 :: 5 : 10, then will 3xX 10 16X5, and 30x80, or x=2 80 And, if : a: bc, then will 3 3ab, or x= 20 *= In like manner, if 12— : ~:: 41, then will 12– 4x 2 2x, and 2x+x=12, orx=4 RULE RULE 7. If any quantity be found on both sides of the equation with the same sign, it may be taken away from them both; and if every term in an equation be multiplied or divided by the same quantity, it may be struck out of them all. b Thus, if 4x+aba, then will 4x=b, and x=-. 4 And, if 3ax+5ab8ac, then will 3x+5b8, and 8c5b 1. Given 5x-15=2x+6; to find the value of *. First, 5x-2x-6+15 Then 3x=21 And 7. 2. Given 40-6x-16—120—14%; to find x. First, 14x-6x=120—40+16 Then 8x=96 And, therefore, x=12 3. Let 5ax-3b-2dx+c be given; to find x. First, 5ax-2dx=c+3b Or 5a-2d Xx=c+36 4. 2 Let 3x-10x=8x+x be given; to find x. 5. Given 6ax3-12abx3-3x2+6ax; to find x. First, dividing the whole by 3ax, we shall have 2x-46=x+2 And then 2x-x=2+46 And then 3x-9+2x-120-3x-57 8. Let ✅✔✅3x+5=7 be given; to find x. First, x7-5=2 And then x22=4 And 2x12, or x=6. TT 9. Les 2 2 And xa Xa2+x*—aa—x3] =aa—2a2x2+x* 4 Or a2x2+x=a^—2a*x*+x* Whence a'x+2a2x2=a^ And consequently x 3a And x 3a2 EXAMPLES FOR PRACTICE. 1. Given x+18=3x-5; to find x. Ans. 11 2. Given 3y-a+b=cd; to find y. Ans. y= 3. Given 6-2x+10=20—3—2; to find x. a 4. Given 3ax+——3—bx—a ; `to find x. cd+a-b. 3 Ans. x=2. 6-3a Ans. x= 6a-2b 7. Given 12+x=2+x; to find x. Ans. x=4. 8. Given 4 8. Given ✓a2+x2=b++x**; to find x. Ans. x 9. Given x+a√/a*+x√b2+x2; to find x. Ans. x 4a REDUCTION OF TWO, THREE, OR MORE, SIMPLE EQUATIONS, CONTAINING TWO, THREE, OR MORE, UNKNOWN QUANTITIES. PROBLEM I. To exterminate two unknown quantities, or to reduce the twa simple equations containing them to one. RULE I. 1. Observe which of the unknown quantities is the least involved, and find its value in each of the equations, by the methods already explained. 2. Let the two values thus found be made equal to each other, and there will arise a new equation with only one unknown quantity in it, whose value may be found as before. |