Write a formula as a sum of fractions with constant numeratorsI'm supposed to write this formula: 9 a + 43 a 2 + 9 a + 20 As a sum of fractions with constant numerators as: 7 a + 5 + 2 a + 4 The first step is of course: 9 a + 43 ( a + 5 ) ( a + 4 ) Now it is possible to write it as a sum using the following method: u a + 5 + v a + 4 u + v = 9 a + 43Which gives me: u = 9 a + 43 − v v = 9 a + 43 − u u = 9 a + 43 − ( 9 a + 43 − u ) u = uBut that doesn't help much knowing that u = u. Therefor my question: how can I get u = 7 and v = 2? Any hints are appreciated.

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Write a formula as a sum of fractions with constant numeratorsI&#039;m supposed to write this formula:            9      a      +      43                      a        2            +      9      a      +      20      As a sum of fractions with constant numerators as:      7          a      +      5        +      2          a      +      4      The first step is of course:            9      a      +      43              (      a      +      5      )      (      a      +      4      )      Now it is possible to write it as a sum using the following method:      u          a      +      5        +      v          a      +      4        u  +  v  =  9  a  +  43Which gives me:  u  =  9  a  +  43  −  v  v  =  9  a  +  43  −  u  u  =  9  a  +  43  −  (  9  a  +  43  −  u  )  u  =  uBut that doesn&#039;t help much knowing that   u  =  u. Therefor my question: how can I get   u  =  7 and   v  =  2? Any hints are appreciated.

Jerry Kidd · Accepted Answer

You want to write             9      a      +      43              (      a      +      5      )      (      a      +      4      )       as fractions with constant numerators which clearly must be of the form       p          a      +      5        +      q          a      +      4      . Getting the &quot;common denominator&quot;,             p      (      a      +      4      )              (      a      +      5      )      (      a      +      4      )        +            q      (      a      +      5      )              (      a      +      5      )      (      a      +      4      )        =            p      a      +      4      p      +      q      a      +      5      q              (      a      +      5      )      (      a      +      4      )       and you want   p  a  +  4  p  +  q  a  +  5  q  =  (  p  +  q  )  a  +  (  4  p  +  5  q  )  =  9  a  +  43 for all a. That means that you want p+q= 9 and 4p+ 5q= 43. Solve those equations for p and q.

Hailee Stout · Accepted Answer

You made a mistake when you wrote this:Now it is possible to write it as a sum using the following method:      u          a      +      5        +      v          a      +      4        u  +  v  =  9  a  +  43This should have been:  u  (  a  +  4  )  +  v  (  a  +  5  )  =  9  a  +  43or,  (  u  +  v  )  a  +  (  4  u  +  5  v  )  =  9  a  +  43Since this is an identity in   a, we have  u  +  v  =  9and  4  u  +  5  v  =  43Equate this to get   u and   v.

Write a formula as a sum of fractions with constant numerators I'm supposed to write this formula:

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