Question:Line L passes through the points \((0, -8)\) and \((1, -2)\) in the coordinate plane.Line M is defined by the...

1. TRANSLATE the problem conditions

• Given information:
  • Line L passes through \((0, -8)\) and \((1, -2)\)
  • Line M: \(\frac{\mathrm{p}}{3}\mathrm{x} - \frac{1}{2}\mathrm{y} = 5\)
  • System has no solution
• What "no solution" means: The lines must be parallel but not identical (different y-intercepts)

2. INFER what we need to find

• For parallel lines: slopes must be equal
• For distinct lines: y-intercepts must be different
• Strategy: Find both slopes, set them equal, solve for p

3. SIMPLIFY to find the equation of line L

• Calculate slope: \(\mathrm{m} = \frac{-2-(-8)}{1-0} = \frac{6}{1} = 6\)
• Using point \((0, -8)\): \(\mathrm{y} = 6\mathrm{x} - 8\)
• Line L has slope 6 and y-intercept -8

4. SIMPLIFY to convert line M to slope-intercept form

• Start with: \(\frac{\mathrm{p}}{3}\mathrm{x} - \frac{1}{2}\mathrm{y} = 5\)
• Isolate y: \(-\frac{1}{2}\mathrm{y} = -\frac{\mathrm{p}}{3}\mathrm{x} + 5\)
• Multiply by -2: \(\mathrm{y} = \frac{2\mathrm{p}}{3}\mathrm{x} - 10\)
• Line M has slope \(\frac{2\mathrm{p}}{3}\) and y-intercept -10

5. APPLY CONSTRAINTS for the no-solution condition

• Set slopes equal: \(6 = \frac{2\mathrm{p}}{3}\)
• Solve for p: \(18 = 2\mathrm{p}\), so \(\mathrm{p} = 9\)
• Verify y-intercepts differ: \(-8 \neq -10\) ✓

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often don't correctly interpret what "no solution" means for a system of linear equations. They might think it means the lines intersect at the origin or that one equation has no solution by itself.

This conceptual confusion leads them to set up incorrect equations or try to solve the system directly, getting stuck and eventually guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when converting line M from standard form to slope-intercept form. Common mistakes include sign errors when moving terms or incorrect fraction arithmetic when solving \(-\frac{1}{2}\mathrm{y} = -\frac{\mathrm{p}}{3}\mathrm{x} + 5\).

This leads to an incorrect slope for line M, which then gives a wrong value for p when set equal to the slope of line L.

The Bottom Line:

This problem requires students to understand the geometric meaning of "no solution" (parallel but distinct lines) and then execute careful algebraic manipulation. The conceptual leap from "no solution" to "equal slopes, different intercepts" is where many students get lost.