REDUCTION OF SIMPLE EQUATIONS. 152. The Reduction of an Equation is the process of finding the value or values of the unknown quantity, or the roots of the equation. 153. A root of an equation is said to be verified, if the two members of the equation prove to be equal after the root has been substituted for the unknown quantity. 154. A simple equation may be reduced, by transforming it in such a manner that the unknown quantity shall stand alone, and constitute one member of the equation; the other member will then express the value of the unknown quantity, or the root of the equation. Let it be required to find the value of x in the equation To verify this value of x, substitute it for x in equation (1); we (5) Reducing each term to its simplest form, we obtain 73+1 = 4+4t; whence, by addition, we have 81 = 81, and the value of x is therefore vcrified. 155. It should here be observed that an equation of the first degree, containing but one unknown quantity, can not have more than one root. For, whatever the equation may be, suppose it to be cleared of fractions, and the unknown terms transposed to the first member, and the known terms to the second. Then if we represent the algebraic sum of the coefficients of x by a, and the second. member by b, the equation will take this general form : Now, if possible, suppose that this equation has two roots, r and r'. Then since every root must satisfy the equation, (140), we shall have, by substituting r and r' successively in (1), But equation (5) is impossible, since, by supposition, r—r' is not zero, and a is not zero. Hence, An equation of the first degree can not have more than one root. 156. From these principles and illustrations we derive the fol lowing RULE. I. If necessary, clear the equation of fractions, and perform all the operations indicated. II. Transpose the unknown terms to the first member and the known terms to the second, and reduce each member to its simplest form, factoring, when necessary, with reference to the unknown. quantity. III. Divide both members by the coefficient of the unknown quantity, and the second member will be the value required, or the root of the equation. The three principal steps in the reduction of a simple equation, containing but one unknown quantity, may be briefly stated as follows: 1st. Clearing of fractions. 2d. Transposing and uniting terms. 3d. Dividing by the coefficient of x. PRACTICAL SUGGESTIONS. There are certain cases in which the preceding rule may be modified, with advantage, by special artifices. 1. When the equation contains similar terms, or fractions having a common denominator, these should be united as far as possible before clearing of fractions. Thus, 2. When the equation contains fractions whose numerators or denominators are polynomial, we may clear the equation of its simpler denominators first, uniting the entire quantities at each step, if possible. Thus, х 3. When the equation contains but a single numerical term, we may simply indicate the multiplication of this term, in the clearing of fractions, until the final step in the reduction is reached. Thus, х Given 4 + + + = 88; multiplying by 84, dividing by 44, 21x+12x+7x+4x=88 × 84; 44 88 X 84; x = 2 × 84; |