Question

determine the value of \(a,b,c\) that satisfy the following system of equations:

\[(a+b)(a+b+c)=7\]
\[(b+c)(a+b+c)=8\]
\[(a+c)(a+b+c)=9\]

Answer 1

First, divide all equations by \(a+b+c\), that will help a bit.

Answer 2

Now, \[
(a+b) - (b+c) = a-c
\]And \[
(a-c) + (a+c) = 2a
\]

Answer 3

So \[
a = \frac{(a+b)-(b+c)+(a+c)}{2} = \frac{7-8+9}{2(a+b+c)} = \frac{4}{a+b+c}
\]It works by symmetry for \(b\) and \(c\). \[
b = \frac{8-9+7}{2(a+b+c)} = \frac{3}{a+b+c} \\
c= \frac{9-7+8}{2(a+b+c)} = \frac{5}{a+b+c}
\]

Answer 4

Now we add them all up: \[
a+b+c = \frac{4+3+5}{a+b+c} = \frac{12}{a+b+c}\implies(a+b+c)^2=12\\
a+b+c = \sqrt{12}
\]

Answer 5

This leads us with: \[
a= \frac{4}{\sqrt{12}},\quad b=\frac{3}{\sqrt{12}},\quad c=\frac{5}{\sqrt{12}}
\]