X. tion expressing the equality of the quantities 4, and 6-x. Also, x-5-0 is an equation which asserts that x-5 is equal to nothing, and therefore that the positive part of the expression is equal to the negative part. (2.) A simple equation is one which, when cleared of fractions and surds, contains only the first power of the unknown quantity. (See Definitions 3 and 4, page 1; and also Definitions 32 to 36, page 4.) (6.) A cubic equation, or an equation of three dimensions, is one into which the cube of the unknown quantity enters, with the simple and quadratic powers. (7.) A pure quadratic is one into which only the square of the unknown quantity enters. (8.) An adfected quadratic is one which involves the square of the unknown quantity, and also the simple power and unknown quantities. Thus, a+b=0 is a pure quadratic, and ax2+bx+ 0 is an adfected quadratic. (9.) The resolution of equations is the determining from some quantities given the values of others which are unknown, so that these latter may answer certain conditions proposed. (10.) And these values are called roots of the equation. (11.) If equal quantities be added to equal quantities, the sums will be equal. (12.) If equal quantities be taken from equal quantities, the remainders will be equal. (13.) If equal quantities be multiplied by the same or equal quantities, the products will be equal. (14.) If equal quantities be divided by the same or equal quantities, the quotients will be equal. (15.) If the same quantity be added to and subtracted from another, the value of the latter will not be altered. (16.) If a quantity be both multiplied and divided by another, its value will not be altered. (17) Any quantity may be transposed from one side of an equation to the other, by changing its sign: Because, in this transposition, the same quantity is merely subtracted from each side of the equation; and if equals be taken from equals, the remainders are equal. Thus, if x+9=15, and 9 be subtracted from each side, x=15-9, or 6. Also, if x+b= a, and b be subtracted from each side,, x And if x-c=d, and c be added to each side,, x=d+c. -b. Also, if 5x-7=2x+2, and 2x be taken from each side, 52- 2x-7=2, or 3x-7=2; and if -7 be subtracted, or (which is the same thing) if +7 be added to each side, 3x=2+7=9. Also, if x-a+b-c-3x, then, by subtracting—a+b—3x from each side, we have x+3x-a-b+c. Cor. 1. Hence, if the signs of all the terms on each side of an equation be changed, the two sides still remain equal; because in this change every term is transposed. Cor. 2. Hence, when the known and unknown quantities are connected in an equation by the signs or, they may be separated by transposing the known quantities to one side, and the unknown to the other. Cor. 3. Hence, also, if any quantity be found on both sides of an equation, it may be taken away from each; thus, if x+y=5 +y, then x 5. If a-b-cd-b, then a=c+d. (18.) If every term on each side of an equation be multiplied by the same quantity, the results will be equal: Because, in multiplying every term on each side by any quantity, the value of the whole side is multiplied by that quantity; and (13) if equals be multiplied by the same quantity, the products will be equal. Thus, if x=5+a, then 6x=30+6a, by multiplying every term by 6. Cor. 1. Hence an equation, of which any part is fractional, may be reduced to an equation expressed in integers, by multiplying every term by the denominator of the fraction. If there be more fractions than one in the given equation, it may be so reduced by multiplying every term either by the product of the denominators, or by a common multiple of them; and if the least common multiple be used, the equation will be in its lowest terms. Thus, if x+x+4x=13; if every term be multiplied by 12, which is the least common multiple of 2, 3, 4; 6x+4x+3x=156. Cor. 2. Hence, also, if every term on both sides have a common multiplier or divisor, that common multiplier or divisor may be taken away; thus, if ax2+abx cdx; each term being divided by the common multiplier x, ax + ab = cd. Also, if x + a+6 4x-7 then also 5x+a+6—4x—7. 4 4 Cor. 3. Also, if each member of the equation have a common divisor, the equation may be reduced by dividing both sides by that common divisor; = Thus, if ax-ax-abx-ab, each side is divisible by ax-a2, whence xb. Cor. 4. Hence also any term of an equation may be made a square, by multiplying all the terms of the equation by the quantities necessary; as, if ax2+bcxcd, the first term may be made a square by multiplying each term by a, and a2x2+ abcx = acd3. (19.) If each side of an equation be raised to the same power, the results are equal; thus, if x=6, x2—36 ; If xa-y-b, then x2+2ax+ay2 2by+b2. And if the same roots be extracted on each side, the results are equal; thus, if x2-49, x-7. If ab3, then xab; if x2+2x+1=y2-y+1, then x+1= y-, and if x2-4ax+4a2=y2+6by+962, then x-2a=y+36. For (13 and 14) when equal quantities on each side of an equation are multiplied or divided by equal quantities, the results will be equal. Cor. Hence, if that side of the equation which contains the unknown quantity be a perfect square, cube, or other power, by extracting the square root, cube root, &c. of both sides, the equation will be reduced to one of lower dimensions: thus, if x2+8x +16=36, x+46 ; If x3-3x2+3x+1=27, x+1=3; If x2+2x3+x2=100, x2+x=10. = b, = (20.) Any equation may be cleared of a single radical quantity by transposing all the other terms to the contrary side, and raising each side to the power denominated by the surd. If there are more than one surd, the operation must be repeated. Thus, if x=√(ax+b2), by squaring each side x-ax+b2, which is free from surds. Also, if (+7)+x=7, then, (17) by transposition, (x2+7)=7-x; and (19) by squaring each side, x2+7= 49-14x+2, which is free from surds. Also, if x+√(a2x) = then, (17) by transposition, (ax) = b-x; and, (19) by cubing each side, ax= a2x - 3 - · 3b2x + 3bx2· 23, which is free from surds. Also, if {√(x2+21)}—1=x, then, (17) by transposition, √ { x2+√(x2+21)}=x+1, and (19) by squaring each side, 22+ √(x2+21)=x2+2x+1; therefore, (17, Cor. 3,) (x2+21)=2x +1, and, (19) by squaring each side, x2+21=4x+4x+1, which is free from surds. And, if {a2x + √(a3x3)}=c, (19) by cubing each side, a2x + (a3x3) = c3, and (17) by transposition, √(a3x3) ==c3—a2x; therefore, (19) by squaring each side, a3x3—c -2a2c3x+ax', which is free from surds. (21.) Any proportion may be converted into an equation; for the product of the extremes is equal to the product of the means. Let a: b::c: d, by the nature of proportion (18, Cor. 1,) ad-bc. a C ; therefore, SIMPLE EQUATIONS. RULE I. Any term may be transposed or transferred from one side of an equation to the other, by changing its signs. Thus, if x+3=7; then will x=7—3, or x=4. And, if x—4—6—8; then will x—8—4—6—6. a+b=c―d; then will x-a-b+c—d. And, if 4x-8-3x+20; then 4x-3x-20+8, and consequently x 28. From this rule it also follows, that if a quantity be found on each side of an equation, with the same sign, it may be left out of both of them; and that the signs of all the terms of any equation may be changed from to, or from to, without altering its value. + Thus, if x+5=7+5; then, by cancelling, x=7. And if a-x-b-c; then, by changing the signs, x-a-c-b, or x=x+c-b. 1. Given 2x+3=x+17 to find x. By transposing, gives 2x-x-17-3-14. 2. Given 5x-9=4x+7 to find x. By transposing, gives 5x-4x-7+9=16. 3. Given x+9-2-4 to find x. Ans. x 14. Whence x=14, Ans. Ans. x=16. Whence x=16, Ans. Ans. x-3. By transposing, gives x-4+2-9--3, Ans. 4. Given 9x-8-8x-5 to find x. By transposing, gives 9x-8x-8-5-3. Ans. x=3. Whence x=3, Ans. 5. Given 7x+8-3-6x+4 to find x. By transposing, gives 7x-6x=4—3—8— —1. -1, Ans. 6. What number is that, to the double of which if 18 be added the sum will be 82 ? Let x= the number required. Then 2x+18=82; position, 2x-64, and 2-32. by trans 7. What number is that, to the double of which if 44 be added, the sum is equal to four times the required number? x Let the number. Then 2x+44-4x; .. by transposition, 44-2x, and 22=x. 8. What number is that, the double of which exceeds its half by 6? Let x the number. Then 2x-1x=6; .. 4x-x=12, or 3x =12, ... x=4. 9. From two towns which are 187 miles distant, two travellers set out at the same time with an intention of meeting. One of them goes 8 miles and the other 9 miles a day. In how many days will they meet? the number of days required; then 8x Let the number of miles one travelled, and 9x the number the other travelled; and since they meet, they must together have travelled the whole distance, consequently 8x+9x=187, or 172-187,.. 11. RULE II. If an unknown quantity has a coefficient, its value may be found by dividing both sides of the equation by that coefficient, because if two equal quantities be divided by the same or equal quantities, the quotients will be equal. And, if 2x+4=16; then will x+2-8, or x-8-2-6. Also, if=5+3; then will x=10+6=16. 2x And, if—2—4; then 2x-6=12, or, by division, x—3—6, or x=9. 1. Given 16x+-2-34 to find x. Ans. 2-2. 32 By transposing, gives 16x-34—2—32, and by division 2-16 =2, Ans. 2. Given 4x-8-3x+13 to find x. Ans. -3. Here 4x-8-3x+13, by transposing, gives 4x+3x=13+8, or 7-21, and, by division, 21 =3, Ans. 7 3. Given 10x-19=7x+17 to find x. Ans. x=12. Here 10x-197x+17, by transposition, 10x-7x=17+19, or 36 3 3x36, and, by division, z =12, Ans. Ans. x-2. 4. Given 8x-3+9=7x+9+27 to find x. Here 8x-3+9=7x+9+27, by transposing, 8x+7x=27+ 30 15 9-9+3, or 15x-30, and, by division, x- =2, Ans. 5. Given 3ax-3ab-12d. 4d Ans. xb a Here 3ax-3ab-12d, by transposition, 3ax-12d+3ab, and, by |