4. Let Vxx be converted into an infinite series. 5. Let 31x3 be converted into an infinite series. AN EQUATION is when two equal quantities, differently expressed, are compared together by means of the sign = placed between them. Thus, 12-5-7 is an equation, expressing the equality of the quantities 12—5 and 7. A simple equation is that, which contains only one unknown quantity, in its simple form, or not raised to any power. Thus, x-a+b=c is a simple equation, containing only the unknown quantity x. Reduction of equations is the method of finding the value of the unknown quantity. It consists in ordering the equation. so, that the unknown quantity may stand alone on one side of the equation without a coefficient, and all the rest, or the known quantities, on the other side. RULE 1.* Any quantity may be transposed from one side of the equation to the other, by changing its sign. * These are founded on the general principle of performing equal operations on equal quantities, when it is evident, that the results must still be equal; whether by equal additions, or subtractions, or multiplications, or divisions, or roots, or powers. Thus, if x+3=7, then will x=7—3=4. And, if x-4+6=8, then will x=8+46=6. Also, if x-a+b=c-d, then will x-c-d+a-b.. And, in like manner, if 4x-8=3x+20, then will 4x-3x =20+8, or x=28. RULE 2. If the unknown term be multiplied by any quantity, that quantity may be taken away by dividing all the other terms of the equation by it. Thus, if ax-ab-a, then will x-b-1. And if 2x+4=16, then will x+2=8, and x=8—2—6. In like manner, if ax+2ba=3c3, then will x+26=. 3c and , a If the unknown term be divided by any quantity, that quantity may be taken away by multiplying all the other terms of the equation by it. Thus, if = 5+3, then will x=10+6=16. 2 And, if ~=b+c—d, then will x=ab+ac-ad. a In like manner, if: 2x -2=6+4, then will 2x-6=18+12, 3 and 2x=18+12+6=36, or x=36=18. RULE 4. The unknown quantity in any equation may be made free from surds by transposing the rest of the terms according to the rule, and then involving each side to such a power, as is denoted by the index of the said surd. Thus, if Vx-2=6, then will √x=6+2=8, and x-8=64. And, if 4x+16=12, then will 4x+16=144, and 4x=144 3 In like manner, if V2x+3+4=8, then will 2x+3=84=4, And 2x+3=43=64, and 2x=64-3=61, or x=1=304. 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 equa tion. Thus, if x2+6x+9=25, then will x+3=25=5, or x=5 -3=2. And, if 3x-9-21+3, then will 3x=21+3+9=33, and x2-33=11, or x =V11. In like manner, if +10=20, then will 2x+30=60, 2x and x2+15=30, or x2=30-15-15, or x=V15. 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 3xx10=16x5, and 30x=80, or x=8=2. 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+a=b+a, then will 4-6, and x= 4 And, if 3ax+5ab=8ac, then will 3x+56=8c, and x= 8c-5b 3 In like manner, if 2=—=—;, then will 2x=16, and x=8. 2x MISCELLANEOUS EXAMPLES. 1. Given 5x-15=2x+6; to find the value of x. First, 5x-2x=6+15 Then 3x=21 And x=31-7. 2. Given 40-6x-16-120-14x; to find x. First, 14x-6x=120-40+16 Then 8.x-96 And, therefore, x==12. 3. Let 5ax-3b=2dx+c be given; to find x. First, 5ax-2dx=c+3b Or 5a-2dxx=c+3b |