Algebraic Manipulation with Simultaneous Equations

Question

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*}

Answer 1

Add the three equations altogether to get:

$(ax+by+c)+(a+bx+cy)+(ay+b+cx)=(ax+bx+cx)+(ay+by+cy)+(a+b+c)= (a+b+c)(x+y+1)=7x+7y+7$

Since $x+y$ is not equal to $-1$, we can divide both sides of the equation by $x+y+1$ to get $a+b+c=(7x+7y+7)/(x+y+1)=7$ which gives us what we wanted.