https://file.aiquickdraw.com/mathbot/file/17273829791335w0r16ca.jpg

Let's go through the parts of the problem.

### Part (a)
The line equation is given as:
\[
3x - 4y = 12
\]
You are given two points that lie on this line: \((0, A)\) and \((B, 0)\).

1. **For the point \((0, A)\):**
   Substitute \(x = 0\) into the equation \(3x - 4y = 12\):
   \[
   3(0) - 4A = 12 \implies -4A = 12 \implies A = -3
   \]
   
2. **For the point \((B, 0)\):**
   Substitute \(y = 0\) into the equation \(3x - 4y = 12\):
   \[
   3B - 4(0) = 12 \implies 3B = 12 \implies B = 4
   \]

So, the values are:
\[
A = -3 \quad \text{and} \quad B = 4
\]

### Part (b)
Now, using the values from part (a), we know the intercepts of the line:
- The line passes through \((0, -3)\) on the \(y\)-axis.
- The line passes through \((4, 0)\) on the \(x\)-axis.

Looking at the graph provided, you need to identify which line has these intercepts. The line that passes through these points is **Line C**.

### Summary of Answers:
- Part (a): \(A = -3\), \(B = 4\)
- Part (b): The correct line is **C**.

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:
1. How would the equation change if the slope were steeper?
2. What happens to the intercepts if the equation changes to \(3x + 4y = 12\)?
3. How can you convert the equation \(3x - 4y = 12\) into slope-intercept form?
4. Can you determine the slope and y-intercept of the line from its equation?
5. How would you plot a line with known intercepts on a graph?

**Tip**: Always substitute given coordinates into the line equation to verify if the points lie on that line.

The coordinates (0, A) and (B, 0) lie on the line 3x - 4y = 12. What are the values of A and B?

Learn how to find the values of A and B when given coordinates on the line equation 3x - 4y = 12. This problem involves determining intercepts and plotting the correct line on the graph. Suitable for students in Grades 8-10.

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