y = 1/2x + 8 y = cx + 10 In the system of equations above, c is a constant....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{y = \frac{1}{2}x + 8}\)
  • \(\mathrm{y = cx + 10}\)
  • The system has no solution
• Need to find: the value of c

2. INFER what "no solution" means

• A system of linear equations has no solution when the lines are parallel
• Parallel lines have the same slope but different y-intercepts
• Since we have different y-intercepts (8 and 10), we need the same slope

3. TRANSLATE each equation to identify slopes and y-intercepts

• Both equations are in slope-intercept form: \(\mathrm{y = mx + b}\)
• First equation: \(\mathrm{y = \frac{1}{2}x + 8}\)
  • Slope = \(\mathrm{\frac{1}{2}}\)
  • Y-intercept = 8
• Second equation: \(\mathrm{y = cx + 10}\)
  • Slope = c
  • Y-intercept = 10

4. INFER the relationship needed

• For no solution: slopes must be equal
• Therefore: \(\mathrm{c = \frac{1}{2}}\)

Answer: \(\mathrm{\frac{1}{2}}\) (also acceptable: 0.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the conditions for "no solution" with "infinite solutions"

Students might think that for no solution, everything must be different, or they might remember that infinite solutions require the same equation (same slope AND same y-intercept). This conceptual confusion leads them to set up incorrect conditions, potentially thinking c should NOT equal \(\mathrm{\frac{1}{2}}\).

This leads to confusion and guessing among the available answer choices.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students correctly identify that slopes should be equal but make arithmetic errors in reading the slope from \(\mathrm{y = \frac{1}{2}x + 8}\)

They might misread \(\mathrm{\frac{1}{2}}\) as 2, or confuse slope with y-intercept, leading them to think c should equal 8 or some other value.

This causes them to get stuck and abandon systematic solution.

The Bottom Line:

This problem tests whether students truly understand what it means for a linear system to have no solution. The key insight is connecting "no solution" to parallel lines, then applying that geometric understanding to the algebraic representations.