To find two numbers $x$ and $y$ such that their difference and the difference of their cubes are equal to two given numbers, we are given the system of equations:

x - y = a \tag{1} x^3 - y^3 = b \tag{2}

Step-by-step derivation:

1. Express $x$ in terms of $y$: From equation (1), we have: x = y + a \tag{3} Substitute this into equation (2).

2. Substitute $x = y + a$ into the cubic equation: We use the identity for the difference of cubes: $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$ Using equation (1) $x - y = a$, we get: x^3 - y^3 = a \cdot (x^2 + xy + y^2) \tag{4} Now substitute $x = y + a$ from equation (3) into this expression for $x^2 + xy + y^2$.

3. Expand $x^2 + xy + y^2$: Substituting $x = y + a$ into the quadratic terms: $x^2 = (y + a)^2 = y^2 + 2ay + a^2$ $xy = (y + a)y = y^2 + ay$ Now sum these up: x^2 + xy + y^2 = (y^2 + 2ay + a^2) + (y^2 + ay) + y^2 = 3y^2 + 3ay + a^2 \tag{5}

4. Substitute back into equation (4): Substituting $x^2 + xy + y^2 = 3y^2 + 3ay + a^2$ into equation (4), we get: $x^3 - y^3 = a \cdot (3y^2 + 3ay + a^2) = b$ Expanding this: 3a y^2 + 3a^2 y + a^3 = b \tag{6}

Necessary conditions for rational solutions:

Equation (6) is a quadratic equation in $y$: 3a y^2 + 3a^2 y + (a^3 - b) = 0 \tag{7} For $y$ to have a rational solution, the discriminant of this quadratic equation must be a perfect square (since rational roots of a quadratic require the discriminant to be a perfect square).

The discriminant $\Delta$ of the quadratic equation $3a y^2 + 3a^2 y + (a^3 - b) = 0$ is given by: $\Delta = (3a^2)^2 - 4 \cdot 3a \cdot (a^3 - b)$ Simplifying: $\Delta = 9a^4 - 12a(a^3 - b)$ $\Delta = 9a^4 - 12a^4 + 12ab$ $\Delta = -3a^4 + 12ab$ Thus, for $y$ to be rational, the discriminant $-3a^4 + 12ab$ must be a perfect square.

Example with $a = 10$ and $b = 2120$:

For $a = 10$ and $b = 2120$, we substitute into the discriminant formula: $\Delta = -3(10)^4 + 12(10)(2120)$ $\Delta = -3(10000) + 12(10)(2120)$ $\Delta = -30000 + 254400$ $\Delta = 224400$ Since $224400$ is a perfect square (it's equal to $474^2$), there exists a rational solution in this case.

Necessary condition:

To ensure a rational solution for $y$, the discriminant $\Delta = -3a^4 + 12ab$ must be a perfect square.

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Would you like to explore any particular step in more detail or need further clarification?

Here are five related questions to expand the discussion:

1. How does the value of $a$ affect the discriminant?
2. Can we solve for specific values of $y$ when the discriminant is a perfect square?
3. What happens if $\Delta$ is not a perfect square?
4. How does this method change if we modify the equation $x^3 - y^3 = b$?
5. How can we generalize this approach for different forms of polynomials?

Tip: Always check the discriminant when solving quadratic equations for rational solutions!