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The given equation is \( 3x - 2y = 12 \), and the graph represents this equation as a straight line. The task is to identify which of the coordinate pairs listed are solutions to this equation.

To solve this, we will check if the coordinates satisfy the equation \( 3x - 2y = 12 \). We do this by plugging in the \( x \)- and \( y \)-values of each point into the equation and verifying if the equality holds.

Let's check each option:

### A: \( (2, -3) \)
\[
3(2) - 2(-3) = 6 + 6 = 12 \quad \text{(True)}
\]
So, \( (2, -3) \) is a solution.

### B: \( (4, 0) \)
\[
3(4) - 2(0) = 12 - 0 = 12 \quad \text{(True)}
\]
So, \( (4, 0) \) is a solution.

### C: \( (5, -1) \)
\[
3(5) - 2(-1) = 15 + 2 = 17 \quad \text{(False)}
\]
So, \( (5, -1) \) is not a solution.

### D: \( (0, -6) \)
\[
3(0) - 2(-6) = 0 + 12 = 12 \quad \text{(True)}
\]
So, \( (0, -6) \) is a solution.

### E: \( (2, 3) \)
\[
3(2) - 2(3) = 6 - 6 = 0 \quad \text{(False)}
\]
So, \( (2, 3) \) is not a solution.

### Conclusion:
The correct answers are **A**, **B**, and **D**.

Would you like further details on this process, or do you have any other questions?

Here are five related questions:
1. How do you determine the slope of the line from the equation \( 3x - 2y = 12 \)?
2. Can you rewrite the equation in slope-intercept form?
3. What is the importance of the intercepts of the line in relation to graphing?
4. How can you quickly check if a point lies on a line without plugging values into the equation?
5. What is the geometric interpretation of the solutions of the equation?

**Tip:** When checking if a point lies on a line, substituting the point’s coordinates into the line's equation can give a quick verification.

Select all coordinate pairs that represent a solution to the equation 3x - 2y = 12.

Explore how to find the coordinate pairs that solve the equation 3x - 2y = 12 by analyzing the graph. This problem involves key algebraic concepts, graphing skills, and identifying solutions in the coordinate plane. Ideal for students in Grades 7-9.

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