What is the slope of the graph of \(\mathrm{y = \frac{1}{4}(27x + 15) + 7x}\) in the xy-plane?

1. INFER the solution strategy

• Given: \(\mathrm{y = \frac{1}{4}(27x + 15) + 7x}\)
• Goal: Find the slope
• Strategy: Convert to slope-intercept form \(\mathrm{y = mx + b}\) where \(\mathrm{m}\) is the slope

2. SIMPLIFY using distributive property

• Distribute \(\mathrm{\frac{1}{4}}\) to both terms in parentheses:
  • \(\mathrm{y = (\frac{1}{4})(27x) + (\frac{1}{4})(15) + 7x}\)
  • \(\mathrm{y = \frac{27x}{4} + \frac{15}{4} + 7x}\)

3. SIMPLIFY by combining like terms

• Convert 7x to fourths: \(\mathrm{7x = \frac{28x}{4}}\)
• Combine x-terms: \(\mathrm{\frac{27x}{4} + \frac{28x}{4} = \frac{(27 + 28)x}{4} = \frac{55x}{4}}\)
• Final form: \(\mathrm{y = \frac{55x}{4} + \frac{15}{4}}\)

4. INFER the slope from standard form

• In \(\mathrm{y = mx + b}\) form, we have \(\mathrm{m = \frac{55}{4}}\) and \(\mathrm{b = \frac{15}{4}}\)
• Therefore, slope = \(\mathrm{\frac{55}{4} = 13.75}\)

Answer: 55/4 or 13.75

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make fraction arithmetic errors when combining like terms. They might incorrectly convert 7x to \(\mathrm{\frac{7x}{4}}\) instead of \(\mathrm{\frac{28x}{4}}\), leading to \(\mathrm{\frac{27x}{4} + \frac{7x}{4} = \frac{34x}{4}}\), giving a slope of \(\mathrm{\frac{34}{4} = 8.5}\). This leads to confusion and an incorrect answer.

Second Most Common Error:

Missing INFER connection: Students may correctly apply the distributive property but fail to recognize they need to combine like terms to get slope-intercept form. They might leave the equation as \(\mathrm{y = \frac{27x}{4} + \frac{15}{4} + 7x}\) and incorrectly identify the slope as just \(\mathrm{\frac{27}{4}}\), missing the additional 7x term entirely.

The Bottom Line:

This problem tests whether students can systematically transform a linear equation into slope-intercept form through careful algebraic manipulation, particularly requiring precise fraction arithmetic when combining terms.