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To derive the equation of a proportional linear relationship using the slope formula, follow these steps:

### Step-by-Step Guide:

1. **Identify the Proportional Relationship**:
   - A proportional relationship between two variables implies that as one variable increases, the other increases at a constant rate. The graph of this relationship is a straight line passing through the origin (0, 0).
   
2. **Slope Formula**: 
   The slope formula is given by:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line.

3. **Choose Points on the Line**:
   Since it is a proportional relationship, one of the points will always be \( (0, 0) \). Let the other point on the line be \( (x_2, y_2) \).

4. **Calculate the Slope (m)**:
   Using \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) \), substitute into the slope formula:
   \[
   m = \frac{y_2 - 0}{x_2 - 0} = \frac{y_2}{x_2}
   \]
   This gives the slope of the line.

5. **Write the Equation**:
   Since the line passes through the origin, the equation of the line is of the form:
   \[
   y = mx
   \]
   Substituting the value of \( m = \frac{y_2}{x_2} \), you get:
   \[
   y = \frac{y_2}{x_2} \cdot x
   \]
   This is the equation for the proportional linear relationship, where \( m = \frac{y_2}{x_2} \) is the constant of proportionality.

Would you like a more detailed breakdown or additional examples?

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Here are some follow-up questions you may want to explore:
1. How does a non-proportional relationship differ from a proportional one?
2. What happens to the slope if you pick different points on the line?
3. Can the slope ever be negative in a proportional relationship?
4. How do you interpret the constant of proportionality in real-world problems?
5. How would you find the slope if the points do not involve the origin?

**Tip**: Always check that the line passes through the origin before concluding it's a proportional relationship.

How can you use the slope formula to derive the equation of a proportional linear relationship?

Learn how to use the slope formula to derive the equation of a proportional linear relationship. This guide covers step-by-step instructions, including identifying a proportional relationship, calculating the slope, and writing the equation of the line in the form y = mx. Suitable for students in Grades 8-10.

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