QUESTION STEM:In the xy-plane, line q is parallel to the line with equation 6y - 9x = 12. Line q...

1. TRANSLATE the given line equation to find its slope

• Given: \(6\mathrm{y} - 9\mathrm{x} = 12\)
• Convert to slope-intercept form \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\):
  • Add \(9\mathrm{x}\) to both sides: \(6\mathrm{y} = 9\mathrm{x} + 12\)
  • Divide everything by 6: \(\mathrm{y} = \frac{9}{6}\mathrm{x} + \frac{12}{6}\)
  • Simplify fractions: \(\mathrm{y} = \frac{3}{2}\mathrm{x} + 2\)
• The slope is \(\mathrm{m} = \frac{3}{2}\)

2. INFER what we know about line q

• Since line q is parallel to the given line, it must have the same slope: \(\frac{3}{2}\)
• Line q passes through its x-intercept \((4, 0)\)
• Strategy: Use point-slope form to write the equation of line q

3. SIMPLIFY to find line q's equation

• Apply point-slope form: \(\mathrm{y} - \mathrm{y_1} = \mathrm{m}(\mathrm{x} - \mathrm{x_1})\)
• Substitute our values: \(\mathrm{y} - 0 = \frac{3}{2}(\mathrm{x} - 4)\)
• Distribute: \(\mathrm{y} = \frac{3}{2}\mathrm{x} - \frac{3}{2}(4)\)
• Calculate: \(\mathrm{y} = \frac{3}{2}\mathrm{x} - 6\)

4. TRANSLATE to find the y-intercept

• In slope-intercept form \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\), the y-intercept is the constant term \(\mathrm{b}\)
• From \(\mathrm{y} = \frac{3}{2}\mathrm{x} - 6\), the y-intercept is \(-6\)
• Verification: When \(\mathrm{x} = 0\), \(\mathrm{y} = \frac{3}{2}(0) - 6 = -6\)

Answer: -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when converting \(6\mathrm{y} - 9\mathrm{x} = 12\) to slope-intercept form, particularly when dividing by 6 or simplifying fractions.

For example, they might get \(\mathrm{y} = \frac{9}{6}\mathrm{x} + 2\) and incorrectly simplify \(\frac{9}{6}\) as \(\frac{3}{2}\), or make errors with the constant term. This leads to using the wrong slope for line q, resulting in an incorrect final answer.

Second Most Common Error:

Poor INFER reasoning: Students correctly find the slope but then get confused about how to use the x-intercept \((4, 0)\) to find the y-intercept.

Some students might try to substitute \(\mathrm{x} = 4\) into the original given line equation instead of using it to write the equation for line q. Others might forget that they need to write a new equation for line q first. This leads to confusion and often results in selecting an incorrect answer or guessing.

The Bottom Line:

This problem tests your ability to work with parallel lines systematically. The key insight is recognizing that you must first extract the slope from the given line, then use that slope along with the x-intercept to build the equation of the new parallel line. Many students rush and skip the step of actually writing line q's equation, which is essential for finding its y-intercept.