A line passes through the origin and the point \((4, 10)\). Another point Q lies on the same line and...

1. TRANSLATE the problem information

• Given information:
  • Line passes through origin \(\mathrm{(0,0)}\) and point \(\mathrm{(4,10)}\)
  • Point Q lies on this same line
  • Q has x-coordinate 2
• We need to find: y-coordinate of Q

2. INFER the solution approach

• Since all points on a line follow the same linear relationship, I need to:
  • First find the equation of the line using the two known points
  • Then use this equation to find the y-coordinate when \(\mathrm{x = 2}\)
• The most efficient way is to find the slope first

3. SIMPLIFY to find the slope

• Using the slope formula with points \(\mathrm{(0,0)}\) and \(\mathrm{(4,10)}\):

\(\mathrm{slope = \frac{10-0}{4-0} = \frac{10}{4} = \frac{5}{2}}\)

4. INFER the line equation

• Since the line passes through the origin, it has no y-intercept
• The equation is: \(\mathrm{y = \frac{5}{2}x}\)

5. SIMPLIFY to find point Q

• Substitute \(\mathrm{x = 2}\) into the equation:

\(\mathrm{y = \frac{5}{2}(2) = 5}\)

Answer: C (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't recognize that they need to find the slope first, or they try to work with the points directly without establishing the linear relationship.

They might attempt to find some pattern between the coordinates (like 4→10, so 2→5) without understanding the underlying mathematical relationship. While this might lead to the correct answer by luck, it shows conceptual confusion about how linear equations work.

This approach might work for this specific problem but would fail on more complex variations, leading to inconsistent problem-solving.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find slope but make arithmetic errors.

Common mistakes include:

• Calculating slope as \(\mathrm{\frac{10}{4} = 2.5}\) instead of recognizing it as \(\mathrm{\frac{5}{2}}\)
• Getting confused with fraction multiplication: \(\mathrm{\frac{5}{2} \times 2 = \frac{5}{2}}\) instead of 5
• Mixing up which coordinate goes where in the slope formula

This leads them to select Choice A (3), Choice B (4), or Choice D (8) depending on their specific calculation error.

The Bottom Line:

This problem tests whether students understand that linear relationships are consistent across all points on a line. The key insight is recognizing that once you know two points on a line, you can find any other point using the slope relationship.