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Let's solve the problem step by step.

### Part a) 
We are given that the points \((0, A)\) and \((B, 0)\) lie on the line \(5x - 3y = 15\). We need to find the values of \(A\) and \(B\).

#### Step 1: Substitute \((0, A)\) into the equation
The point \((0, A)\) lies on the line, so we substitute \(x = 0\) and \(y = A\) into the equation \(5x - 3y = 15\):
\[
5(0) - 3A = 15 \implies -3A = 15 \implies A = -5
\]

#### Step 2: Substitute \((B, 0)\) into the equation
Next, the point \((B, 0)\) also lies on the line, so substitute \(x = B\) and \(y = 0\) into the equation \(5x - 3y = 15\):
\[
5B - 3(0) = 15 \implies 5B = 15 \implies B = 3
\]

Thus, the values are:
\[
A = -5 \quad \text{and} \quad B = 3
\]

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### Part b)
We are asked to use the answers from part a) to determine which line on the graph represents \(5x - 3y = 15\).

#### Step 1: Plot the points
We know two points on the line: \((0, -5)\) and \((3, 0)\). These points should allow us to identify the line visually on the graph.

Looking at the graph, the magenta line (labeled **E**) passes through the points \((0, -5)\) and \((3, 0)\). Therefore, the line \(5x - 3y = 15\) corresponds to line **E** on the graph.

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### Final Answer:
- Part a: \(A = -5\), \(B = 3\)
- Part b: The line representing \(5x - 3y = 15\) is line **E**.

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Would you like further explanation on any part of this solution?

Here are 5 related questions to expand your understanding:
1. How do we find the equation of a line given two points?
2. What is the slope of the line \(5x - 3y = 15\) and how does it relate to the graph?
3. Can you describe how to solve the equation \(ax + by = c\) for different values of \(x\) and \(y\)?
4. How do we find the intercepts of a line algebraically and graphically?
5. How would this method change if we had a non-linear equation, such as a quadratic?

**Tip:** When solving for coordinates on a line, using intercepts (where the line crosses the axes) is an efficient method to quickly plot or identify a line.

The coordinates (0, A) and (B, 0) lie on the line 5x - 3y = 15. What are the values of A and B? Use your answer to work out which line on the graph represents 5x - 3y = 15.

This math problem involves finding the coordinates (0, A) and (B, 0) on the line 5x - 3y = 15 and using them to identify the corresponding line on a graph. The solution covers algebraic concepts like linear equations and coordinate geometry, suitable for students in Grades 8-10.

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