Two lines are defined by the equations below, where k is a constant.Line 1: 2/3x - 1/4y = 1/2Line 2:...

1. TRANSLATE the problem information

• Given information:
  • Line 1: \(\frac{2}{3}\mathrm{x} - \frac{1}{4}\mathrm{y} = \frac{1}{2}\)
  • Line 2: \(\mathrm{kx} + \mathrm{y} = -2\)
  • Need to find k where lines coincide

2. INFER the approach for coincident lines

• For two lines to coincide (be the same line), one equation must be a scalar multiple of the other
• Strategy: Manipulate Line 1 to match the form of Line 2, then compare coefficients

3. SIMPLIFY Line 1 to match Line 2's form

• Line 2 has the form: (coefficient)x + y = -2
• To get Line 1 in this form, multiply the entire equation by -4:

\(-4(\frac{2}{3}\mathrm{x} - \frac{1}{4}\mathrm{y}) = -4(\frac{1}{2})\)

• Distribute the -4:

\(-4 \times (\frac{2}{3})\mathrm{x} - 4 \times (-\frac{1}{4})\mathrm{y} = -4 \times (\frac{1}{2})\)

\(-\frac{8}{3}\mathrm{x} + \mathrm{y} = -2\)

4. INFER the value of k by comparing equations

• Line 1 rearranged: \(-\frac{8}{3}\mathrm{x} + \mathrm{y} = -2\)
• Line 2: \(\mathrm{kx} + \mathrm{y} = -2\)
• Comparing coefficients of x: \(\mathrm{k} = -\frac{8}{3}\)

Answer: A (\(-\frac{8}{3}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that coincident lines require one equation to be a scalar multiple of the other. Instead, they might try to solve the system as if looking for intersection points, leading to confusion when they realize the system should have infinite solutions rather than one solution.

This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors or fraction calculation mistakes when multiplying Line 1 by -4. A common error is getting the signs wrong, such as getting \(+\frac{8}{3}\) instead of \(-\frac{8}{3}\), or miscalculating the fraction arithmetic.

This may lead them to select Choice D (\(\frac{3}{2}\)) or Choice C (\(-\frac{2}{3}\)).

The Bottom Line:

This problem requires recognizing the conceptual relationship between coincident lines and scalar multiples, then executing careful algebraic manipulation with fractions. Students often struggle with the strategic insight more than the calculations.