A system of equations is given by y = ax + 5/2 and 3y - 9x = 3/2. The constant...

1. TRANSLATE the second equation to slope-intercept form

• Given: \(\mathrm{3y - 9x = \frac{3}{2}}\)
• Convert to \(\mathrm{y = mx + b}\) form:
  • Add 9x to both sides: \(\mathrm{3y = 9x + \frac{3}{2}}\)
  • Divide by 3: \(\mathrm{y = 3x + \frac{1}{2}}\)

2. INFER the conditions for no solution

• We now have two lines:
  • Line 1: \(\mathrm{y = ax + \frac{5}{2}}\)
  • Line 2: \(\mathrm{y = 3x + \frac{1}{2}}\)
• Key insight: A system has no solution when lines are parallel but distinct
• This means: same slope, different y-intercepts

3. APPLY CONSTRAINTS to find the value of a

• For parallel lines: slopes must be equal
  • Slope of Line 1 = \(\mathrm{a}\)
  • Slope of Line 2 = \(\mathrm{3}\)
  • Therefore: \(\mathrm{a = 3}\)
• Verify distinct lines: y-intercepts must be different
  • Y-intercept of Line 1 = \(\mathrm{\frac{5}{2}}\)
  • Y-intercept of Line 2 = \(\mathrm{\frac{1}{2}}\)
  • Since \(\mathrm{\frac{5}{2} ≠ \frac{1}{2}}\) ✓, lines are indeed distinct

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing "no solution" with "different slopes"

Students may think that no solution means the lines don't intersect because they have completely different slopes, not realizing that parallel lines (same slope, different y-intercepts) are what create no solution scenarios. This leads to setting \(\mathrm{a ≠ 3}\) or creating an equation that makes the slopes different, causing confusion and potentially guessing random values.

Second Most Common Error:

Poor TRANSLATE execution: Algebraic errors when converting to slope-intercept form

Students might make mistakes like forgetting to divide all terms by 3, or sign errors when rearranging \(\mathrm{3y - 9x = \frac{3}{2}}\). For example, getting \(\mathrm{y = 3x - \frac{1}{2}}\) instead of \(\mathrm{y = 3x + \frac{1}{2}}\). This gives an incorrect slope or y-intercept, leading to a wrong value for a.

The Bottom Line:

This problem tests whether students understand that "no solution" in a system means parallel but distinct lines, not just "different" lines. The algebraic manipulation is straightforward, but the conceptual understanding of what creates an inconsistent system is the real challenge.