The relationship between two variables, a and b, is linear. For every increase in the value of a by 3,...

1. TRANSLATE the rate of change information

• Given information:
  • Linear relationship between variables a and b
  • For every increase of 3 in a, b decreases by 12
  • Point: when a = 5, b = 7

• What this tells us: We have a rate of change and one specific point on the line.

2. INFER the slope from the rate of change

• The slope represents how much b changes for each unit change in a
• We know: a increases by 3 → b decreases by 12
• Therefore: \(\mathrm{slope = \frac{change\,in\,b}{change\,in\,a} = \frac{-12}{3} = -4}\)

3. INFER the appropriate equation form to use

• We have: \(\mathrm{slope = -4}\) and point \(\mathrm{(5, 7)}\)
• With a point and slope, point-slope form works perfectly: \(\mathrm{b - b_1 = m(a - a_1)}\)

4. SIMPLIFY using point-slope form

• Substitute our values: \(\mathrm{b - 7 = -4(a - 5)}\)
• Expand: \(\mathrm{b - 7 = -4a + 20}\)
• Solve for b: \(\mathrm{b = -4a + 20 + 7 = -4a + 27}\)

5. SIMPLIFY to convert to standard form

• Current form: \(\mathrm{b = -4a + 27}\)
• Move all variables to left side: \(\mathrm{4a + b = 27}\)

Answer: A (\(\mathrm{4a + b = 27}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the rate description "for every increase in a by 3, b decreases by 12" as meaning the slope is \(\mathrm{-12/3}\) but calculate it as \(\mathrm{12/3 = 4}\), missing the negative sign from "decreases."

This creates the equation \(\mathrm{b = 4a - 13}\) instead of \(\mathrm{b = -4a + 27}\), which converts to standard form as \(\mathrm{4a - b = 13}\). This may lead them to select Choice C (\(\mathrm{4a + b = 13}\)) after making additional sign errors.

Second Most Common Error:

Poor INFER reasoning: Students correctly find \(\mathrm{slope = -4}\) but struggle with choosing the right approach. Some attempt to use slope-intercept form directly without using the given point \(\mathrm{(5, 7)}\), leading to incorrect y-intercept values.

This confusion about method selection causes them to get stuck and guess among the answer choices.

The Bottom Line:

This problem requires precise translation of English rate descriptions into mathematical slope calculations, combined with systematic application of point-slope form. The key insight is recognizing that "decreases by 12" means the slope contribution is negative.