What is the slope of the graph of \(\mathrm{y = \frac{1}{3}(29x + 10) + 5x}\) in the xy-plane?

1. INFER the strategy needed

• Given: \(\mathrm{y = \frac{1}{3}(29x + 10) + 5x}\)
• Goal: Find the slope
• Key insight: The slope is the coefficient of x when the equation is in \(\mathrm{y = mx + b}\) form
• Strategy: Distribute and combine like terms to get standard form

2. SIMPLIFY by distributing the fraction

• Apply distributive property to \(\mathrm{\frac{1}{3}(29x + 10)}\):

\(\mathrm{y = \frac{29}{3}x + \frac{10}{3} + 5x}\)

3. SIMPLIFY by combining like terms

• Convert 5x to have denominator 3: \(\mathrm{5x = \frac{15}{3}x}\)
• Combine the x terms:

\(\mathrm{y = \frac{29}{3}x + \frac{15}{3}x + \frac{10}{3}}\)

\(\mathrm{y = (\frac{29}{3} + \frac{15}{3})x + \frac{10}{3}}\)

\(\mathrm{y = \frac{44}{3}x + \frac{10}{3}}\)

4. INFER the final answer

• The equation is now in \(\mathrm{y = mx + b}\) form where \(\mathrm{m = \frac{44}{3}}\) and \(\mathrm{b = \frac{10}{3}}\)
• Therefore, the slope is \(\mathrm{\frac{44}{3}}\)

Answer: \(\mathrm{\frac{44}{3}}\) (which equals approximately 14.67)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making arithmetic errors when combining fractions like \(\mathrm{\frac{29}{3} + \frac{15}{3}}\)

Students might incorrectly compute this as \(\mathrm{\frac{29}{3} + \frac{5}{3} = \frac{34}{3}}\) (forgetting that \(\mathrm{5 = \frac{15}{3}}\), not \(\mathrm{\frac{5}{3}}\)). Or they might add numerators and denominators separately: \(\mathrm{\frac{29 + 15}{3 + 3} = \frac{44}{6}}\), then fail to simplify properly.

This leads to confusion and an incorrect slope value, causing them to guess among the answer choices.

The Bottom Line:

This problem tests algebraic fluency more than conceptual understanding. Students who can distribute and combine like terms accurately will succeed, while those with weak fraction arithmetic skills will struggle despite understanding the slope concept.