A line in the xy-plane has a slope of 2/3 and passes through the point \((-6, 0)\). Which equation represents...

Part 1: Brief Solution

This tests understanding of linear equations in slope-intercept form and the ability to translate given information into mathematical expressions.

Key steps:

• Identify slope-intercept form: \(\mathrm{y = mx + b}\)
• Use given slope: \(\mathrm{m = \frac{1}{9}}\)
• Recognize point \(\mathrm{(0, 14)}\) provides y-intercept: \(\mathrm{b = 14}\)
• Substitute to get: \(\mathrm{y = \frac{1}{9}x + 14}\)

Answer: B

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Sign Confusion on Y-intercept - Phase: Executing Approach → Choice A (\(\mathrm{y = \frac{1}{9}x - 14}\))
   • Process skill failure: Translate
   • Students incorrectly use -14 instead of +14 for the y-intercept despite the point being \(\mathrm{(0, 14)}\).
2. Slope Sign Error - Phase: Devising Approach → Choice D (\(\mathrm{y = -\frac{1}{9}x + 14}\))
   • Process skill failure: Translate
   • Students mistakenly make the slope negative, writing \(\mathrm{-\frac{1}{9}}\) instead of the given positive \(\mathrm{\frac{1}{9}}\).
3. Double Sign Error - Phase: Executing Approach → Choice C (\(\mathrm{y = -\frac{1}{9}x - 14}\))
   • Process skill failure: Translate
   • Students make both slope and y-intercept negative due to general confusion about signs in linear equations.

Part 3: Detailed Solution

To find the equation of a line, we use the slope-intercept form: \(\mathrm{y = mx + b}\), where m is the slope and b is the y-intercept.

Process Skill: TRANSLATE - We must convert the English description into mathematical components. The problem states 'slope of 1/9,' which means \(\mathrm{m = \frac{1}{9}}\). The phrase 'passes through the point (0, 14)' gives us a specific coordinate pair that the line must satisfy.

Here's the crucial insight: Process Skill: INFER - The point \(\mathrm{(0, 14)}\) has x-coordinate 0, which means it lies directly on the y-axis. This point IS the y-intercept! When a line passes through \(\mathrm{(0, y_0)}\), that \(\mathrm{y_0}\) value becomes our b-value in the slope-intercept form.

Since the point is \(\mathrm{(0, 14)}\), our y-intercept \(\mathrm{b = 14}\).

Substituting into slope-intercept form:

\(\mathrm{y = mx + b}\)
\(\mathrm{y = \frac{1}{9}x + 14}\)

Let's verify this works by checking our given point:

When \(\mathrm{x = 0}\):

\(\mathrm{y = \frac{1}{9}(0) + 14}\)
\(\mathrm{y = 0 + 14}\)
\(\mathrm{y = 14}\) ✓

This confirms point \(\mathrm{(0, 14)}\) lies on our line, validating our equation.

Answer: B) \(\mathrm{y = \frac{1}{9}x + 14}\)

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

• Slope Sign Misinterpretation: Students might assume the slope should be negative without justification, leading to choice D. This represents a Translate process skill failure where they don't properly convert the clearly stated 'slope of 1/9' into the positive value \(\mathrm{\frac{1}{9}}\).

Errors while executing the approach:

• Y-intercept Sign Confusion: The most common error is writing -14 instead of +14 (choice A). This Translate failure occurs when students don't recognize that point \(\mathrm{(0, 14)}\) means when \(\mathrm{x = 0}\), \(\mathrm{y = 14}\), requiring a positive 14 in the equation.
• Compound Sign Errors: Some students make both slope and y-intercept negative (choice C), representing multiple Translate failures where they systematically misinterpret the given information.
• Point Substitution Errors: Students might incorrectly think that passing through \(\mathrm{(0, 14)}\) means subtracting 14, not understanding that the y-intercept form requires addition of the b-value.

Errors while selecting the answer:

• Verification Skip: Students fail to substitute the given point back into their chosen equation to verify correctness, missing opportunities to catch sign errors before finalizing their answer.