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Real numbers $a$, $b$, $c$, $x$, and $y$ satisfy
\begin{align*}
ax+by+c&=x+7,\\
a+bx+cy&=2x+6y-4,\\
ay+b+cx&=8x+5y-3,
\end{align*}
and $x + y \neq -1$.  Find $a+b+c$.

 Sep 9, 2023
 #1
avatar+128398 
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\( $x + y \neq -1$ \)

 

Add the equations

 

a ( x + y) + a  + b(x + y) + b + c (x + y) + c  = 11x + 11y 

 

(a + b + c)  + (x + y) (a + b + c)  = 11 ( x + y)

 

(a + b + c) (x + y + 1)  = 11 (x + y)

 

a + b + c  =   11 (x + y)  / ( x + y + 1) 

 

cool cool cool

 Sep 10, 2023

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