What is the slope of the graph of y = 5x/13 - 23 in the xy-plane?

1. TRANSLATE the problem information

• Given: \(\mathrm{y = \frac{5x}{13} - 23}\)
• Need to find: the slope of this line

2. INFER the approach needed

• To find slope, I need to identify the slope-intercept form: \(\mathrm{y = mx + b}\)
• The slope is the coefficient of x (the value m)
• I should rewrite the equation to clearly see this form

3. TRANSLATE the equation into standard slope-intercept form

• \(\mathrm{y = \frac{5x}{13} - 23}\) can be written as:
• \(\mathrm{y = \frac{5}{13}x + (-23)}\)
• This matches \(\mathrm{y = mx + b}\) where:
  • \(\mathrm{m = \frac{5}{13}}\) (slope)
  • \(\mathrm{b = -23}\) (y-intercept)

4. SIMPLIFY to get final answer

• The slope is \(\mathrm{\frac{5}{13}}\)
• If decimal form is needed: \(\mathrm{5 ÷ 13 = 0.3846}\) (use calculator)

Answer: \(\mathrm{\frac{5}{13}}\) or 0.384 or 0.385 or .3846

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which part of the equation represents the slope versus the y-intercept. They might incorrectly identify -23 as the slope since it's the more obvious number, leading them to answer -23 or 23.

Second Most Common Error:

Missing conceptual knowledge of slope-intercept form: Students who don't recognize \(\mathrm{y = mx + b}\) may not know how to systematically extract the slope. This leads to confusion and guessing among any numerical values they see in the equation.

The Bottom Line:

This problem tests whether students can recognize and work with the fundamental form of linear equations. The fractional coefficient makes it slightly more complex than basic examples, but the core skill is pattern recognition of \(\mathrm{y = mx + b}\).