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 $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$
+324 Set the 2 equations equal to each other. You will get
x³+8y³=x+8y.
Try to solve this now.
x + 8y =7
x = 7 - 8y sub this into the 2nd equation
(7 - 8y)^3 + 8y^3 = 7, solve for y
Solve for y:
(7 - 8 y)^3 + 8 y^3 = 7
Expand out terms of the left hand side:
-504 y^3 + 1344 y^2 - 1176 y + 343 = 7
Subtract 7 from both sides:
-504 y^3 + 1344 y^2 - 1176 y + 336 = 0
The left hand side factors into a product with three terms:
-168 (y - 1)^2 (3 y - 2) = 0
Divide both sides by -168:
(y - 1)^2 (3 y - 2) = 0
Split into two equations:
(y - 1)^2 = 0 or 3 y - 2 = 0
Take the square root of both sides:
y - 1 = 0 or 3 y - 2 = 0
Add 1 to both sides:
y = 1 or 3 y - 2 = 0
Add 2 to both sides:
y = 1 or 3 y = 2
Divide both sides by 3:
y = 1 or y = 2/3 sub these two values into the 1st equation to get:
x = - 1 or x =5/3
