y = 6x + 18 One of the equations in a system of two linear equations is given. The system...

1. TRANSLATE the problem requirements

• Given: \(\mathrm{y = 6x + 18}\) (first equation)
• Need: Second equation that creates a system with no solution
• What this means: We need parallel but distinct lines

2. INFER the key relationship

• For no solution: Lines must be parallel (same slope) AND distinct (different y-intercepts)
• The given equation has \(\mathrm{slope = 6}\) and \(\mathrm{y\text{-intercept} = 18}\)
• We need another line with \(\mathrm{slope = 6}\) but \(\mathrm{y\text{-intercept} ≠ 18}\)

3. TRANSLATE each answer choice to slope-intercept form

• A. \(\mathrm{-6x + y = 18}\) → \(\mathrm{y = 6x + 18}\) (same as given equation)
• B. \(\mathrm{-6x + y = 22}\) → \(\mathrm{y = 6x + 22}\)
• C. \(\mathrm{-12x + y = 36}\) → \(\mathrm{y = 12x + 36}\)
• D. \(\mathrm{-12x + y = 18}\) → \(\mathrm{y = 12x + 18}\)

4. INFER which creates no solution

• Choice A: Identical equations → infinitely many solutions
• Choice B: Same slope (6), different y-intercept (22 vs 18) → no solution ✓
• Choices C & D: Different slopes (12 vs 6) → exactly one solution

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse "no solution" with "same equation" and think identical equations create no solution.

They might reason: "If the equations are exactly the same, then there's no new information, so no solution." This leads them to select Choice A (\(\mathrm{-6x + y = 18}\)) without recognizing that identical equations actually mean every point on the line satisfies both equations (infinitely many solutions).

Second Most Common Error:

Poor TRANSLATE execution: Students make algebraic errors when converting between equation forms, particularly with signs.

For example, they might incorrectly convert \(\mathrm{-6x + y = 22}\) to \(\mathrm{y = -6x + 22}\) instead of \(\mathrm{y = 6x + 22}\), leading them to think the slopes are different. This causes confusion about which lines are actually parallel, leading to guessing among the choices.

The Bottom Line:

This problem tests understanding of what "no solution" actually means in systems of equations - students must connect the abstract concept to the concrete geometric relationship of parallel but distinct lines.