If 
3
x
−
y
=
12
, what is the value of 
8
x
2
y
?

A) 
2
12

B) 
4
4

C) 
8
2

D) The value cannot be determined from the information given.

To solve for the value of \(8x^2y\), given the equation \(3x - y = 12\), let's analyze the situation:

1. **Rearrange the given equation**:
   \[
   y = 3x - 12
   \]
   This expresses \(y\) in terms of \(x\).

2. **Substitute into \(8x^2y\)**:
   Substitute \(y = 3x - 12\) into the expression \(8x^2y\):
   \[
   8x^2y = 8x^2(3x - 12)
   \]

3. **Distribute and simplify**:
   \[
   8x^2y = 8x^2(3x) - 8x^2(12) = 24x^3 - 96x^2
   \]

The value of \(8x^2y\) depends on the specific value of \(x\), which is not provided. Therefore, the exact numerical value cannot be determined without additional information about \(x\).

### The correct answer is:
**D) The value cannot be determined from the information given.**

Would you like to dive deeper into any part of this explanation or have any questions?

Here are 8 questions related to the topic:

1. How do you isolate \(x\) in the equation \(3x - y = 12\)?
2. What happens to the expression \(8x^2y\) if you substitute different values of \(x\)?
3. How can you determine if an equation has a unique solution?
4. Can the value of \(8x^2y\) be determined if \(x = 2\)?
5. What would change if the equation was \(3x + y = 12\) instead?
6. How do you solve quadratic equations that arise from similar problems?
7. What methods can you use to check the validity of the given solutions?
8. How does the graph of the equation \(3x - y = 12\) look like?

**Tip:** Always ensure all necessary variables are defined before attempting to find a specific numerical value.

Learn how to determine the value of 8x^2y when given the equation 3x - y = 12. Understand the use of substitution and distribution in algebra to solve expressions involving variables. Suitable for students in Grades 7-9.

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