16. + y x+y y (x2-y2)' c+ a (b−c) (c− a) * (c− a) (a —b) * (a−b) (b −c) ° 2xy 20. 1 1 (n+1)(n+2) (n + 1) (n + 2) (n+3) (n + 1) (n + 3) * 21. a2-bc b2-ac c2-ab + From this we learn that we may change the sign of the denominator of a fraction if we also change the sign of the numerator. Hence if the numerator or denominator, or both, be expressions with more than one term, we may change the sign of every term in the denominator if we also change the sign of every term in the numerator. or, writing the terms of the new fraction so that the positive terms may stand first, 182. Again, since = a-x ax+x2+5αx - x2 бах ab the product of a and b, and ab= the product of +a and b, the sign of a product will be changed by changing the signs of one of the factors composing the product. Hence (a-b) (b-c) will give a set of terms, and (b-a) (b-c) will give the same set of terms with different signs. This may be seen by actual multiplication: (a−b)(b−c)=ab — ac—b2+bc, (b-a)(b−c)=-ab+ac+b2-bc. Consequently if we have a fraction 1 (a-b) (b-c)' and we change the factor a-b into b-a, we shall in effect change the sign of every term of the expression which would result from the multiplication of (a—b) into (b−c). Now we may change the signs of the denominator if we also change the signs of the numerator (Art. 180); If we change the signs of two factors in a denominator, the sign of the numerator will remain unaltered, thus (a−b)(b−c) † (b−a) (a−c) — (c—a) (c—b) • First change the signs of the factor (b-a) in the second fraction, changing also the sign of the numerator; and change the signs of the factor (c-a) in the third fraction, changing also the sign of the numerator, the result is 1 -1 + Next, change the signs of the factor (c-b) in the third, changing also the sign of the numerator, = the result is (a−b)(b-c) (a−b) (a–c) (a–c) (b−c)* L.C. M. of the three denominators is (a−b) (b−c) (a−c), a-c + -b+c (a-b)(b-c)(a–c) (a−b) (a−c)(b-c) a-b (a-b)(a–c)(b−c) 5. + + (m-2) (m—3) (m-1) (3-m) (m −1)(m −2) * (a−b)(x+b) ̄ (b−a)(x+a)* 10. a(a−b)(a−c) +8 (b−a) (b−c) *c(c—a) (e—b)• 184. Ex. To simplify 1 2-11x+30 1 Here the denominators may be expressed in factors, and we have The L. C. M. of the denominators is (x − 5) (x − 6) (x − 7), and we have (x-5) (x−6) (x-7) (x-5)(x−6) (x−7) 9. 1−x+x2- Ñ3 + XII. ON FRACTIONAL EQUATIONS. 185. We shall explain in this Chapter the method of solving, first, equations in which fractional terms occur, and secondly, Problems leading to such equations. 186. An equation involving fractional terms may be reduced to an equivalent equation without fractions by multiplying every term of the equation by the Lowest Common Multiple of the denominators of the fractional terms, This process is in accordance with the principle laid down in Ax. III. page 58, for if both sides of an equation be multiplied by the same expression, the resulting products will, by that Axiom, be equal to each other. 187. The following examples will illustrate the process of clearing an Equation of Fractions. + =14x-28, |