What is the slope of the graph of y = (26x + 15)/3 + 4x in the xy-plane?

1. INFER the solution strategy

• Given: \(\mathrm{y = \frac{26x + 15}{3} + 4x}\)
• Goal: Find the slope
• Key insight: We need slope-intercept form \(\mathrm{y = mx + b}\) where the slope is the coefficient of x

2. SIMPLIFY by distributing the fraction

• Distribute the division across the numerator:
  \(\mathrm{y = \frac{26x + 15}{3} + 4x}\)
  \(\mathrm{y = \frac{26x}{3} + \frac{15}{3} + 4x}\)
• Simplify the constant: \(\mathrm{\frac{15}{3} = 5}\)
  \(\mathrm{y = \frac{26x}{3} + 5 + 4x}\)

3. SIMPLIFY by combining like terms

• Convert 4x to thirds so we can add the x-terms:
  \(\mathrm{4x = \frac{12x}{3}}\)
• Now combine the x-terms:
  \(\mathrm{y = \frac{26x}{3} + 5 + \frac{12x}{3}}\)
  \(\mathrm{y = \frac{26x + 12x}{3} + 5}\)
  \(\mathrm{y = \frac{38x}{3} + 5}\)

4. INFER the final answer

• The equation is now in slope-intercept form: \(\mathrm{y = mx + b}\)
• The slope \(\mathrm{m = \frac{38}{3}}\)

Answer: A) \(\mathrm{\frac{38}{3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make fraction arithmetic errors, especially when converting 4x to twelfths for combining terms.

Many students might incorrectly think \(\mathrm{4x = \frac{4x}{3}}\) instead of \(\mathrm{\frac{12x}{3}}\), leading to:
\(\mathrm{y = \frac{26x + 4x}{3} + 5 = \frac{30x}{3} + 5 = 10x + 5}\)

This gives them a slope of 10, which would lead them to select Choice B (12) as the closest answer, or cause confusion since 10 isn't listed.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly start the simplification but stop before combining all like terms.

They might get to \(\mathrm{y = \frac{26x}{3} + 5 + 4x}\) and think this is the final form, trying to identify the slope as either \(\mathrm{\frac{26}{3}}\) or 4. This leads them to select Choice C (\(\mathrm{\frac{26}{3}}\)) since it appears directly in their partial simplification.

The Bottom Line:

This problem tests careful algebraic manipulation more than conceptual understanding. Students who rush through the fraction arithmetic or don't fully combine like terms will miss the correct slope value, even if they understand that slope-intercept form is the goal.