If $a,b,c$ satisfy the system of equations \begin{align*}b + c &= 12-3a \\ a+c &= -14 - 3b \\ a+b &= 7 - 3c, \end{align*}

Adding all the equations produces what we need

2a +  2b  + 2c  = -3 (a + b + c)  +  5 

5a + 5b + 5c  =   5

(2/5) [ 5a +  5b + 5c ]  =   (2/5) 5

2a + 2b +2c   = 2

2a + 2b + 2c = 6