To find the ordered pairs (x, y) that are solutions to the equation 3x + 5y = 205y + x^2, we first need to rearrange the equation into a standard form.
Starting from:
3x + 5y = 205y + x2
we can move all terms to one side:
x2 - 3x + 200y = 0
This looks like a quadratic equation in terms of x. To find the values of y that yield integer solutions for x, we can use the discriminant of this quadratic equation. The discriminant is given by the formula:
D = b2 – 4ac
For our equation:
- a = 1
- b = -3
- c = 200y
Now substituting these into the discriminant formula:
D = (-3)2 - 4(1)(200y) = 9 - 800y
For the quadratic to have real solutions, the discriminant must be greater than or equal to zero:
9 - 800y 6 0
Solving for y:
800y 9
y rac{9}{800}
This means y can take on values equal to or less than 0.01125. However, we are often looking for integer values, so the only integer value for y that fulfills this condition is:
- y = 0
Now, substituting y = 0 back into our rearranged equation to solve for x gives:
x2 - 3x = 0
Factoring out x:
x(x - 3) = 0
This results in:
- x = 0
- x = 3
Thus, we can form the following ordered pairs that satisfy the original equation:
- (0, 0)
- (3, 0)
In conclusion, the values that create ordered pairs that are solutions to the equation are (0, 0) and (3, 0).