x3 = 12x y−3 + = −6The solution to the given system of equationsis x y( , ). What is the value of y ?A) −3B) 6C) 18D) 30

To solve the given system of equations for $y$, we will follow the steps outlined.

1. Break Down the Problem

We have the following equations:

1. $x^3 = 12x$
2. $y - 3 = -6$

We need to find the value of $y$ after determining $x$.

2. Relevant Concepts

We will solve the first equation for $x$ and then use the second equation to find $y$.

3. Analysis and Detail

Step 1: Solve for $x$
From the first equation: $x^3 - 12x = 0$ Factoring gives: $x(x^2 - 12) = 0$ Thus, $x = 0$ or $x^2 - 12 = 0$. This leads to: $x^2 = 12 \implies x = \sqrt{12} \text{ or } x = -\sqrt{12} \implies x = 2\sqrt{3} \text{ or } x = -2\sqrt{3}$

Step 2: Solve for $y$
Using the second equation, $y - 3 = -6$: $y = -6 + 3 = -3$

4. Verify and Summarize

Now we have $y = -3$ regardless of the value of $x$. Thus, we don't need to derive the specific $x$ values since $y$ is independently determined.

Final Answer

The value of $y$ is $\boxed{-3}$.