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Algebra Level 2

We are given the system of equation in real numbers x , y x,y :

{ x y + x + y + 7 = 0 3 x + 3 y = 9 + 2 x y \begin{cases} xy+x+y+7=0\\ 3x+3y=9+2xy\\ \end{cases}

Determine x 2 y + x y 2 x^2y+xy^2 .


The answer is 6.

Tom Zhou by Tom Zhou , 21, Canada

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1 solution

Aquilino Madeira
May 14, 2016

{ x y + x + y + 7 = 0 3 x + 3 y = 9 + 2 x y ⇔ { x + y = − x y − 7 3 ( x + y ) = 9 + 2 x y ⇔ { x + y = − x y − 7 3 ( − x y − 7 ) = 9 + 2 x y { x + y = − x y − 7 − 3 x y − 21 = 9 + 2 x y ⇔ { x + y = − x y − 7 − 5 x y = 30 ⇔ { x + y = − x y − 7 x y = − 6 { x + y = − ( − 6 ) − 7 x y = − 6 ⇔ { x + y = − 1 x y = − 6 x 2 y + x y 2 = x y ( x + y ) = − 6 ( − 1 ) = 6 \begin{array}{l} \left\{ \begin{array}{l} xy + x + y + 7 = 0\\ 3x + 3y = 9 + 2xy \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x + y = - xy - 7\\ 3\left( {x + y} \right) = 9 + 2xy \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x + y = - xy - 7\\ 3\left( { - xy - 7} \right) = 9 + 2xy \end{array} \right.\\ \left\{ \begin{array}{l} x + y = - xy - 7\\ - 3xy - 21 = 9 + 2xy \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x + y = - xy - 7\\ - 5xy = 30 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x + y = - xy - 7\\ xy = - 6 \end{array} \right.\\ \left\{ \begin{array}{l} x + y = - ( - 6) - 7\\ xy = - 6 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x + y = - 1\\ xy = - 6 \end{array} \right.\\ \\ {x^2}y + x{y^2} = xy\left( {x + y} \right) = - 6( - 1) = 6 \end{array}

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