a b + a + b = 250 b c + b + c = 300 a c + a + c = 216then find a + b + c = ?

Question

a  b  +  a  +  b  =  250  b  c  +  b  +  c  =  300  a  c  +  a  +  c  =  216then find   a  +  b  +  c  =  ?

Jenna Farmer · Accepted Answer

{                            a          b          +          a          +          b          =          250                                      b          c          +          b          +          c          =          300                                      a          c          +          a          +          c          =          216                        it seems of no difficulty: a system of 3 equations and 3 variables... Let&#039;s start by adding 1 to every equations, we obtain:      {                            a          b          +          a          +          b          +          1          =          (          a          +          1          )          (          b          +          1          )          =          251                                      b          c          +          b          +          c          +          1          =          (          c          +          1          )          (          b          +          1          )          =          301                                      a          c          +          a          +          c          +          1          =          (          a          +          1          )          (          c          +          1          )          =          217                        Now let&#039;s substitute   a  +  1  =  x  ,  b  +  1  =  y  ,  c  +  1  =  z, so we can reduce the amount of calculus needed, in fact to solve      {                            x          y          =          251                                      y          z          =          301                                      x          z          =          217                        you only need to find   x (or   y) in the first equation, substitute it in the last (or second) equation, and then do another substitution. Then subtract 1, and you&#039;ll find the solutions:  (  a  ,  b  ,  c  )  =  (  −  1  −            7781      43        ,  −  1  −            10793      31        ,  −  1  −  7            1333      251        )and  (  a  ,  b  ,  c  )  =  (            7781      43        −  1  ,            10793      31        −  1  ,  7            1333      251        −  1  )

a b + a + b = 250 b c + b + c = 300 a

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