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Find $a+b+c$, given that $x+y\neq -1$ and
$\begin{align*} ax+by+c&=x+7,\\ a+bx+cy&=2x+6y,\\ ay+b+cx&=4x+y. \end{align*}$

$\small{ \begin{array}{|lrcll|} \hline (1): & ax+by+c &=& x+7 \\ (2): & a+bx+cy &=& 2x+6y \\ (3): & ay+b+cx &=& 4x+y \\ \hline (1)+(2)+(3): & (ax+by+c) \\ & +(a+bx+cy) \\ & +(ay+b+cx) &=& (x+7) \\ &&& +(2x+6y) +(4x+y) \\ & a(1+x+y) \\ & +b(1+x+y)\\ & +c(1+x+y) &=& 7x+7y+7 \\ & (a+b+c)(1+x+y) &=& 7(1+x+y) \quad | \quad :(1+x+y),~ \mathbf{x+y\neq -1} \\ & \mathbf{a+b+c} &=& \mathbf{7} \\ \hline \end{array} }$