The relationship between two variables, x and y, is linear. For each 3-unit increase in x, y increases by 12...

1. TRANSLATE the problem information

• Given information:
  • The relationship is linear
  • For every 3-unit increase in x, y increases by 12 units
  • When \(\mathrm{x = 1}\), \(\mathrm{y = 5}\)
• This tells us we have a rate of change and one specific point on the line

2. INFER the approach

• Since we have a linear relationship, we need the equation \(\mathrm{y = mx + b}\)
• The rate information gives us the slope (m)
• The specific point will help us find the y-intercept (b)
• Strategy: Find slope first, then use the point to find b

3. TRANSLATE the rate information into slope

• Rate of change = change in y ÷ change in x
• Slope = \(\mathrm{12 ÷ 3 = 4}\)
• So our equation becomes: \(\mathrm{y = 4x + b}\)

4. SIMPLIFY to find the y-intercept

• Use the point \(\mathrm{(1, 5)}\) in our equation \(\mathrm{y = 4x + b}\)
• Substitute: \(\mathrm{5 = 4(1) + b}\)
• Simplify: \(\mathrm{5 = 4 + b}\)
• Solve: \(\mathrm{b = 1}\)

5. Write the final equation

• \(\mathrm{y = 4x + 1}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students reverse the rate calculation and compute slope as \(\mathrm{3/12 = 1/4}\) instead of \(\mathrm{12/3 = 4}\)

They might think "3 units of x per 12 units of y" rather than understanding that slope represents "change in y per change in x." This leads them to the equation \(\mathrm{y = (1/4)x + b}\), and after finding b, they don't arrive at any of the given choices, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when solving for the y-intercept

After correctly finding \(\mathrm{slope = 4}\), they substitute the point \(\mathrm{(1, 5)}\) but make errors like:

• \(\mathrm{5 = 4 + b → b = 5 + 4 = 9}\) (adding instead of subtracting)
• This might lead them toward Choice A (\(\mathrm{y = 12x - 7}\)) if they also confuse other aspects of the calculation

The Bottom Line:

This problem tests whether students can correctly interpret rate language and systematically apply the slope-intercept form. The key insight is recognizing that "12 units increase in y for 3 units increase in x" means \(\mathrm{slope = 12/3}\), not 3/12.