ALGEBRAICAL PROBLEMS. SECT. I. DEFINITIONS. (1.) AN Equation is a proposition, which declares the equality of two quantities, expressed algebraically. This is done by connecting these quantities by the sign (=); thus, x 46 x is an equation 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. (3.) A Quadratic Equation, or an equation of two dimensions, is one into which the square of the unknown quantity enters, with or without the simple power. (4.) A Cubic Equation, or an equation of three dimensions, is one into which the cube of the unknown quantity enters, with or without the simple and quadratic powers. (5.) In general, the index of the highest power of the unknown quantity denotes the number of dimensions of the equation. (6.) A Pure Quadratic is one into which only the square of the unknown quantity enters. A (7.) An Adfected Quadratic is one which involves the square of the unknown quantity, and also the simple power and known quantities. Thus, ax2 + b = 0 is a pure quadratic, and ax2 + bx + c = 0 is an adfected quadratic. (8.) 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. (9.) And these values are called Roots of the Equation. (10.) Known quantities are usually expressed by the first letters of the alphabet, a, b, c, &c.; and unknown quantities by the last, v, x, y, &c. And this must be always understood, unless the contrary be expressed. AXIOMS. (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 (12) if equals be taken from equals, the remainders are equal. Thus, if x+9=15, and 9 be subtracted from each side, a 15-9, or 6. tracted from each side, ra- b. c be added to each side, r = d + c. And if x-c=d, and Also, if 5x-7=2x+2, and 2r be taken from each side, 5x-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 -ab-3x from each side, we have r+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 a = c + d. - b = c + d − b, then (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. 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+abr = cdx; each term being divided by the common multiplier x, ax + ab = cd. 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; ax Thus, if ax2 - a2x = abx - a3b, each side is divisible by a2, whence x = b. - 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+bcx=cd, the first term may be made a square by multiplying each term by a, and a2x2 +abcx = ac d2. (19.) If each side of an equation be raised to the same power, the results are equal; = Thus, if x 6, x2 = 36; if x + a = yb, then x2+2ax + a2 = y3 — 2by + b2; 2 |