real numbers x and y satisfy
\(x+xy^2= 250y\)
\(x-xy^2= -240y\)
enter all possible values of x, separated by commas.
Substituting thse values in shows that they do not all work....
desmos graphically show 0,0 35, 7 -35, -7
https://www.desmos.com/calculator/tfuqvjljxr
Add the two equations and you get 2x = 10y, so x = 5y.
Substitute that into the top equation and you get
\(\displaystyle 5y + 5y^{3}=250y, \text{ from which,}\\5y^{3}-245y=0,\\5y(y^{2}-49)=0, \text{ so }y = 0\text{ or }y=\pm7,\\ \text{with corresponding x values}\\x=0\text{ or }x=\pm35.\)
Add the equations and we get that
2x = 10y
x = 5y
So....we have that
5y + 5y^3 = 250y
y^3 + y - 50y =0
y^3 - 49y = 0 factor
y ( y^2 - 49) =0
y( y - 7) (y + 7) = 0
Set each factor to 0 and solve for y and we get that y = 0 y = 7 and y = -7
So x = 0 , 35 and -35
So
(x,y) = (0,0) (7,35) (-7 - 35)