To identify the values that create ordered pairs satisfying the equation 3x + 5y = 205y + x2, we first need to rearrange the equation into a more standard form.
Starting with the original equation:
3x + 5y = 205y + x2
We can move all terms to one side of the equation:
x2 – 3x + 200y = 0
This is a quadratic equation in terms of x. To find values of y for which this equation has real solutions for x, we can use the discriminant (the part of the quadratic formula under the square root) to ensure that the solutions for x are real numbers.
The standard form of a quadratic equation is:
ax2 + bx + c = 0
Where:
- a = 1 (from x2)
- b = -3 (from -3x)
- c = 200y (from 200y)
The discriminant is given by:
D = b2 – 4ac
Plugging our values into this formula gives:
D = (-3)2 – 4(1)(200y)
D = 9 – 800y
For the quadratic to have real solutions, the discriminant must be greater than or equal to zero:
9 – 800y >= 0
Rearranging this inequality:
800y <= 9
y <= 0.01125
This means that for values of y that are less than or equal to 0.01125, we can find corresponding x values that create ordered pairs (x, y) which are solutions to the original equation.
To find specific ordered pairs, we could choose any real number less than or equal to 0.01125 for y and solve for x:
- For example, if we let y = 0, then:
x2 – 3x = 0
x(x – 3) = 0
This gives us x = 0 or x = 3. Thus, ordered pairs (0, 0) and (3, 0) are solutions.
By continuing this process with different values of y below 0.01125, we can generate other ordered pairs that satisfy the equation. For instance:
- For y = 0.01: solve x2 – 3x + 2 = 0
- For y = 0.005: solve x2 – 3x + 1 = 0
The key here is to find values for y that adhere to the inequality we derived, which in turn allows us to calculate the respective x values. Thus, you can create infinite ordered pairs (x, y) as long as you respect the conditions set by the discriminant!