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Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the real solutions to the system \begin{align*} x + 8y &= 7, \\ x^3 + 8y^3 &= 7. \end{align*}Enter $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$

 Dec 5, 2020
 #1
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Let \((x_1,y_1),\ (x_2,y_2),\ \dots \ ,\ (x_n,y_n)\)
be the real solutions to the system
\(\begin{align*} x + 8y &= 7, \\ x^3 + 8y^3 &= 7. \end{align*}\)

Enter \(x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n\).

 

Use Vieta:

 

\(\begin{array}{|rcll|} \hline \mathbf{x^3 + 8y^3} &=& \mathbf{7} \quad | \quad x = 7-8y \\ \left( 7-8y \right)^3 + 8y^3 &=& 7 \\ 7^3- 3*7^2*8y+ 3*7*(8y)^2 - (8y)^3 + 8y^3 &=& 7 \\ \ldots \\ y^3*(8-8^3) + y^2*(3*7*8^2) - y*(3*7^2*8) + 7^3 - 7 &=& 0 \\ y^3*8*(1-8^2) + y^2*(3*7*8^2) - y*(3*7^2*8) + 7^3 - 7 &=& 0 \\ y^3*(-8*63) + y^2*(3*7*8^2) - y*(3*7^2*8) + 7^3 - 7 &=& 0 \quad | \quad : (-8*63) \\\\ y^3 - y^2*\dfrac{3*7*8^2} {8*63} + y*\dfrac{3*7^2*8} {8*63} -\dfrac{7^3 - 7} {8*63} &=& 0 \quad | \quad : (-8*63) \\\\ y^3 - y^2*\dfrac{3*7*8} {3*21} + y*\dfrac{3*7^2*8} {8*63} -\dfrac{7^3 - 7} {8*63} &=& 0 \quad | \quad : (-8*63) \\\\ y^3 - y^2*\dfrac{7*8} {21} + y*\dfrac{3*7^2*8} {8*63} -\dfrac{7^3 - 7} {8*63} &=& 0 \quad | \quad : (-8*63) \\\\ y^3 - y^2*\dfrac{56} {21} + y*\dfrac{3*7^2*8} {8*63} -\dfrac{7^3 - 7} {8*63} &=& 0 \quad | \quad : (-8*63) \\\\ \hline \mathbf{y_1+y_2+y_3} &=& \mathbf{\dfrac{56} {21}} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{x^3 + 8y^3} &=& \mathbf{7} \quad | \quad y = \dfrac{7-x}{8} \\\\ x^3 + 8*\left( \dfrac{7-x}{8} \right)^3 &=& 7 \\\\ x^3 + \dfrac{(7-x)^3}{8^2} &=& 7 \quad | \quad *8^2 \\\\ 8^2x^3 + (7-x)^3 &=& 7*8^2 \\ 8^2x^3 + (7-x)^3 - 7*8^2 &=& 0 \\ \ldots \\ x^3*(8^2-1) + x^2*(3*7) - x*3*7^2 +7^2 - 7*8^2 &=& 0 \\ x^3*63 + x^2*(3*7) - x*3*7^2 +7^2 - 7*8^2 &=& 0 \quad | \quad : 63 \\\\ x^3 + x^2*\dfrac{3*7}{63} - \dfrac{x*3*7^2}{63} + \dfrac{7^2 - 7*8^2}{63} &=& 0 \\\\ x^3 + x^2*\dfrac{3*7}{3*21} - \dfrac{x*3*7^2}{63} + \dfrac{7^2 - 7*8^2}{63} &=& 0 \\\\ x^3 + x^2*\dfrac{7}{21} - \dfrac{x*3*7^2}{63} + \dfrac{7^2 - 7*8^2}{63} &=& 0 \\\\ \hline \mathbf{x_1+x_2+x_3} &=& -\mathbf{\dfrac{7} {21}} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{x_1 + y_1 + x_2 + y_2 + x_3 + y_3} &=& \dfrac{56} {21} - \dfrac{7} {21} \\\\ &=& \dfrac{56-7} {21} \\\\ &=& \dfrac{49} {21} \\\\ \mathbf{x_1 + y_1 + x_2 + y_2 + x_3 + y_3} &=& \mathbf{\dfrac{7} {3}} \\ \hline \end{array}\)

 

laugh

 Dec 5, 2020

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