3. Clear the equation 2x 3xx 74+3=6 of fractions. Ans. 40x-105x+28x=840. x х X 4. Clear the equation +4+ 10 of fractions. 138. An equation may always be cleared of fractions by multiplying each member into all the denominators; but sometimes the same result may be attained by a less amount of multiplication. Thus, in the last example, the equation may be cleared of fractions by multiplying each term by 12 instead of 6×4×2, and it is important to avoid all useless multiplication. In general, an equation may be cleared of fractions by multiplying each member by the least common multiple of all the denominators. 2x 3x 7 5. Clear the equation + of fractions. The least common multiple of all the denominators is 20. If we multiply each member of the equation by 20, we obtain 8x+15x=14. The operation is effected by dividing the least common multiple by each of the denominators, and then multiplying the corresponding numerator, dropping the denominator. It should be remembered that when a fraction has the minus sign before it, this indicates that the fraction is to be subtracted, and the signs of the terms derived from its numerator must be changed, Art. 118. Ans. 36x-3x+12=1. 139. Solution of Equations.—An equation of the first degree containing but one unknown quantity may be solved by transforming it in such a manner that the unknown quantity shall stand alone, constituting one member of an equation; the other member will then denote the value of the unknown quantity. Let it be required to find the value of x in the equation 4х-2. 5х 3х + 5 +5. 8 4 Clearing of fractions, we have 32x-16+25x=30x+200. By transposition we obtain 32x+25x-30x=200+16. Uniting similar terms, 27x=216. Dividing each member by 27, according to Art. 134, we have x=8. To verify this value of x, substitute it for x in the original equation, and we shall have an identical equation, which proves that we have found the correct value of x. 140. Hence we deduce the following RULE. 1. Clear the equation of fractions, and perform all the operations indicated. 2. Transpose all the terms containing the unknown quantity to one side, and all the remaining terms to the other side of the equation, and reduce each member to its most simple form.. 3. Divide each member by the coefficient of the unknown quantity. There are various artifices which may sometimes be em ployed, by which the labor of solving an equation may be considerably abridged. These artifices can not always be reduced to general rules. If, however, any reductions can be made before clearing of fractions, it is generally best to make them; and if the equation contains several denominators, it is often best to multiply by the simpler denominators first, and then to effect any reductions which may be possible before getting rid of the remaining denominators. Sometimes considerable labor may be saved by simply indicating a multiplication during the first steps of the reduction, as we can thus more readily detect the presence of common factors (if there are any), which may be canceled. The discovery of these artifices will prove one of the most useful exercises to the pupil. To verify this result, put 6 in the place of x in the original equation. Solve the following equations: 2. 3ax-4ab=2ax-6ac. 3. 3x2—10x=8x+x2. a (d2+x2)=ac+ ax d 2x+6 7. 3x+ =5+ 5 11x-37 Ans. x=7. 141. A problem in Algebra is a question proposed requiring us to determine the value of one or more unknown quantities from given conditions. 142. The solution of a problem is the process of finding the value of the unknown quantity or quantities that will satisfy the given conditions. 143. The solution of a problem consists of two parts: 1st. The statement, which consists in expressing the conditions of the problem algebraically; that is, in translating the conditions of the problem from common into algebraic language, or forming the equation. 2d. The solution of the equation. The second operation has already been explained, but the first is often more embarrassing to beginners than the second. Sometimes the conditions of a problem are expressed in a distinct and formal manner, and sometimes they are only implied, or are left to be inferred from other conditions. The former are called explicit conditions, and the latter implicit conditions. 144. It is impossible to give a general rule which will enable |