Real numbers x and y satisfy
x + xy^2 = 250y,
x - xy^2 = -240y.
Enter all possible values of x, separated by commas.
first, just add the equations and you get
2x = 10y
x = 5y
now plug that in, and you have
5y + 5y^3 = 250y
5y^3 - 245y = 0
5y(y^2 - 49) = 0
y = -7, 0, 7
and x = 5y
now finish it off
To solve this system of equations, we can add the two equations together to eliminate the term with y^2:
(x + xy^2) + (x - xy^2) = 250y + (-240y)
This simplifies to:
2x = 10y
Now, we can solve for x by isolating it:
x = 10y / 2
Simplifying further:
x = 5y
Therefore, the possible values of x can be any multiple of y, such that y is a real number.
In other words, x can take any value of the form 5y, where y is a real number.
So, the possible values of x are all real numbers of the form 5y, where y is a real number.
To find the values of x that satisfy the given equations, we can use the method of eliminating variables by adding the two equations together. Let's perform the addition:
(x + xy^2) + (x - xy^2) = (250y) + (-240y)
On the left side, the xy^2 terms cancel out, and on the right side, the y terms cancel out:
2x = 10y
Now, we can solve for x by isolating the variable:
2x = 10y
x = 5y
Therefore, any value of x that satisfies the given equations can be expressed as x = 5y, where y is any real number.
To find all possible values of x, we can substitute different values of y into the equation x = 5y and observe the corresponding values of x.
For example:
- If we choose y = 0, then x = 5(0) = 0.
- If we choose y = 1, then x = 5(1) = 5.
- If we choose y = -1, then x = 5(-1) = -5.
As we can see, when y takes different real number values, the corresponding x values are different. Therefore, there are infinitely many possible values of x that satisfy the given equations.
In summary, all possible values of x satisfying the given equations can be expressed as x = 5y, where y is any real number.