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If $a+b+ab=80$, $b+c+bc=120$ and $c+a+ac=24$ then what is $ab+bc+ca$?

I tried substitution to obtain $b$, but then I realized it'd be a quadratic equation which has two values. Likewise, $a$ and $c$ would have two values. So, $ab+bc+ca$ would have more than one result. But my problem specifically asks for "the" solution (which I don't know). I think there's some algebraic way to arrive at the expression. Please help.

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  • $\begingroup$ Are $a,b,c$ supposed to be positive ? $\endgroup$
    – Claude Leibovici
    Aug 15, 2018 at 5:05
  • $\begingroup$ $a, b, c$ are real. $\endgroup$
    – Trdp
    Aug 15, 2018 at 5:07

1 Answer 1

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Hint:

$$(a+1)(b+1)=?$$ etc.

$(80+1)(120+1)(24+1)=$

Alternatively,

choose $a+1=x\iff a=?$ etc.

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