A line passes through the origin and the point \((3, 12)\). A point P moves along this line so that...

1. TRANSLATE the problem information

• Given information:
  • Line passes through origin \((0, 0)\) and point \((3, 12)\)
  • Point P moves along this line
  • x-coordinate increases by 7 units \((\Delta\mathrm{x} = 7)\)
  • Need to find: increase in y-coordinate \((\Delta\mathrm{y} = ?)\)

2. INFER the key relationship

• Since we have two points on the line, we can find its slope
• Slope tells us the rate of change: how much y changes for each unit change in x
• The relationship we need is: \(\Delta\mathrm{y} = \mathrm{slope} \times \Delta\mathrm{x}\)

3. SIMPLIFY to find the slope

• Using slope formula with points \((0, 0)\) and \((3, 12)\):
  \(\mathrm{slope} = \frac{12 - 0}{3 - 0}\)
  \(= \frac{12}{3}\)
  \(= 4\)

4. SIMPLIFY to find the y-coordinate change

• Apply the rate of change relationship:
  \(\Delta\mathrm{y} = \mathrm{slope} \times \Delta\mathrm{x}\)
  \(= 4 \times 7\)
  \(= 28\)

Answer: D (28)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that slope represents the rate of change between coordinates.

Students might calculate the slope correctly \((\mathrm{m} = 4)\) but then get confused about what to do next. They might think they need the actual coordinates of point P, or try to set up a complex equation instead of using the simple relationship \(\Delta\mathrm{y} = \mathrm{m} \cdot \Delta\mathrm{x}\). This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Misinterpreting what "increase by 7 units" means in the context.

Some students might think the x-coordinate becomes 7 (instead of increasing by 7), or confuse which coordinate is changing. They might also mix up the given point \((3, 12)\) with the change in coordinates. This may lead them to select Choice A (7) by thinking the change in y equals the change in x.

The Bottom Line:

This problem tests whether students understand slope as a rate of change, not just as a calculation. The key insight is recognizing that once you know the slope, you can immediately find how much y changes when x changes by any amount.