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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*}

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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*}

Mar 30, 2018

#1
+7348
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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*}

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(ax + by + c) + (a + bx + cy) + (ay + b + cx)  =  (ax + by + c) + (a + bx + cy) + (ay + b + cx)

(ax + by + c) + (a + bx + cy) + (ay + b + cx)  =  (x + 7) + (2x + 6y) + (4x + y)

ax + by + c + a + bx + cy + ay + b + cx  =  x + 7 + 2x + 6y + 4x + y

ax + bx + cx + ay + by + cy + a + b + c  =  7x + 7y + 7

x(a + b + c) + y(a + b + c) + 1(a + b + c)  =  7(x + y + 1)

(a + b + c)(x + y + 1)  =  7( x + y + 1)

a + b + c  =  7

Mar 30, 2018
edited by hectictar  Mar 30, 2018

#1
+7348
+2

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*}

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(ax + by + c) + (a + bx + cy) + (ay + b + cx)  =  (ax + by + c) + (a + bx + cy) + (ay + b + cx)

(ax + by + c) + (a + bx + cy) + (ay + b + cx)  =  (x + 7) + (2x + 6y) + (4x + y)

ax + by + c + a + bx + cy + ay + b + cx  =  x + 7 + 2x + 6y + 4x + y

ax + bx + cx + ay + by + cy + a + b + c  =  7x + 7y + 7

x(a + b + c) + y(a + b + c) + 1(a + b + c)  =  7(x + y + 1)

(a + b + c)(x + y + 1)  =  7( x + y + 1)

a + b + c  =  7

hectictar Mar 30, 2018
edited by hectictar  Mar 30, 2018