
When fractions occur we can sometimes transform the equation to one that does not involve fractions. Example Find the solution to the equation (4 x/ 5)  (7 / 4) = ( x/ 5) + ( x/ 4) . Solution The least common multiple of the denominators in the equation is 4 × 5 = 20 and we proceed as follows: 16 x  35 = 4 x + 5 x 16 x  35 = 9 x adding 35 to both sides and subtracting 9 x from both sides leads to 7 x = 35 so x = 5 is the solution to the equation . Exercise Find the solution to each of the following equations. (a) 5 x  6( x  5) = 2( x + 5) + 5( x  4) (b) ( x + 15)( x  3)  ( x  6 x + 9) = 30  15( x  1) (c) ( x  2) / 2 + ( x + 10) / 9 = 5 Solution (a) 5 x  6( x  5) = 2( x + 5) + 5( x  4) 5 x  6 x + 30 = 2 x + 10 + 5 x  20  x + 30 = 7 x  10 30 = x + 7 x  10 30 = 8 x  10 30 + 10 = 8 x 8 x = 40 x = 5 (b) First, using FOIL , we expand ( x + 15)( x  3) = x  3 x + 15 x  45 = x + 12 x  45 Now we have ( x + 15)( x  3)  ( x14 2  6 x + 9) = 30  15( x  1) x + 12 x  45  x + 6 x  9 = 30  15 x + 15 18 x  54 = 45  15 x 18 x + 15 x  54 = 45 33 x  54 = 45 33 x = 45 + 54 33 x = 99 x = 3 (c) This time we multiply both sides by 2 × 9 . 9( x  2) + 2( x + 10) = 90 9 x  18 + 2 x + 20 = 90 11 x + 2 = 90 11 x = 88 x = 8 Quiz Which of the following is the solution to the equation (x  4)/7 = (x  10)/5 ? (a) 11 (b)  10 (c) 19 (d) 25 Solution: The highest common factor of the denominators is 5 × 7 = 35 . Multiplying both sides of the equation by this 5( x  4) = 7( x  10) 5 x  20 = 7 x  70 5 x  20 + 70 = 7 x  70 + 70 5 x + 50 = 7 x 50 = 7 x  5 x = 2 x x = 25 so that x = 25 is the solution. This can be checked by putting this value into the original equation and showing that each side will have the value 3 . 