c+d 4. Let 7 b + be reduced to an improper frac tion. ANS. 11ab+c+d 2 a In the last two examples, as the value of the frac c+d tion in one, and of in the other, is to be 2 a 2a. added to 7 b, none of the signs are changed. 7. Reduce a b+14- to an improper fraction. 3 m α За 10. Reduce x += to an improper fraction. As these fractions have a common denominator, we evidently obtain the true answer by adding together their numerators, which are similar quantities; and 3 a + 2a = 5 a. For, if a = 4, and c = 2, then 2 a с 8 a =2, or 4; and 3 = 12, or 6; and 4 + 6 = 10 : C 5, or 10, also α 3a+2b -y' x-y 2y-x-3x ANS. a+b. C 5. What is the sum of and /? Although these fractions have a common denominator, still, as the numerators are dissimilar, they can only be added by means of the sign +. Let a = 4, 4 6 b = 6, and c = 2; then (~+) = (+) 2 + 3, or 5; also ato 6. What is the sum of a =1+6=10, or 5. 3b 3 a 7. Add together 2, 3, 4 and. = c+25 x+7=x 10. What is the sum of and? ANS. These fractions having different denominators, they can only be added, as they are proposed, by means of the sign+; we, therefore, reduce them to a common denominator, and proceed as before. From these examples and observations, we derive the following RULE for adding algebraic fractions: Reduce the given fractions to a common denominator ; add their numerators together; and place the sum, for a new numerator, over the common denominator. 5a X bx c d = 5 a b c d, bc 3 x b c X c d = 3 b c cd, the numerators. b We obtain the values of the fractions given in the last column, by dividing all the terms by bc. The fractions are thus reduced to their LEAST common de 16. What is the sum of 6 x, 7, 5, and 3x+2? 17. Add **, 7x+1, 4 x 2 b 3x+5 and 2x+y together. 19. Add together a b, 15, + and. 20. Add together—,—, 4 x and. 21. Add together, andx. It is evident that 3 1 2 x expresses the same value as 3 x and that is equivalent to; and +3 3 3x 22. Add together, and 3x, and 2x, and -x. 23. What is the sum of a, 24. Add together ab, a, anda? 3 Subtraction is performed like addition; but the signs of the quantities to be subtracted must all be changed. |