We are solving the system of equations:

1. $x^3 + y = 0$
2. $3x^2 + y = 0$

Step 1: Solve for $y$ in terms of $x$ from the second equation

From the second equation: $y = -3x^2$

Step 2: Substitute $y = -3x^2$ into the first equation

Substitute $y = -3x^2$ into $x^3 + y = 0$: $x^3 + (-3x^2) = 0$ $x^3 - 3x^2 = 0$

Step 3: Factorize the equation

$x^2(x - 3) = 0$

Step 4: Solve for $x$

The solutions to this equation are: $x^2 = 0 \quad \text{or} \quad x - 3 = 0$ $x = 0 \quad \text{or} \quad x = 3$

Step 5: Solve for $y$ corresponding to each $x$

For $x = 0$: $y = -3(0)^2 = 0$ Solution: $(0, 0)$

For $x = 3$: $y = -3(3)^2 = -3(9) = -27$ Solution: $(3, -27)$

Step 6: Write the solution set

The solution set is: $\{(0, 0), (3, -27)\}$

Final Answer:

The correct choice is: A. The solution set is $\{(0, 0), (3, -27)\}$.

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Would you like a detailed explanation of any step? Here are some related questions:

1. How can the substitution method be applied to other systems of equations?
2. What does the solution set represent graphically for this system?
3. How do we verify solutions for a system of nonlinear equations?
4. Can this system be solved using other methods, such as elimination?
5. What happens if the system involves inequalities instead of equalities?

Tip: When solving systems, always check all potential solutions by substituting them back into the original equations.