ALGEBRAICAL PROBLEMS. SECTION 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— 4 = 6 − x is an equation expressing the equality of the quantities 4 and 6 — x. Also x - 5=0 is an equation which asserts that 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. B (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. |