Consider the system of equations below:3x + ky = 156x - 8y = k + 10For what value of k...

1. TRANSLATE the problem requirement

• Given: System of equations with parameter k
• Find: Value of k that makes the system have no solution
• Key insight: "No solution" is a specific mathematical condition

2. INFER what "no solution" means geometrically

• A system has no solution when the equations represent parallel lines
• Parallel lines: same slope, different y-intercepts
• This creates an inconsistent system (contradictory equations)

3. INFER the coefficient ratio strategy

• For parallel lines, coefficient ratios must be equal
• System: \(\mathrm{3x + ky = 15}\) and \(\mathrm{6x - 8y = k + 10}\)
• x-coefficient ratio: \(\mathrm{\frac{3}{6} = \frac{1}{2}}\)
• y-coefficient ratio must equal: \(\mathrm{\frac{k}{-8} = \frac{1}{2}}\)

4. SIMPLIFY to solve for k

• Set up the proportion: \(\mathrm{\frac{k}{-8} = \frac{1}{2}}\)
• Cross multiply: \(\mathrm{k = (-8) \times \frac{1}{2}}\)
• Calculate: \(\mathrm{k = -4}\)

5. INFER verification is needed

• Substitute \(\mathrm{k = -4}\) back into original system
• Equation 1: \(\mathrm{3x - 4y = 15}\)
• Equation 2: \(\mathrm{6x - 8y = 6}\), which simplifies to \(\mathrm{3x - 4y = 3}\)
• Since \(\mathrm{3x - 4y}\) cannot equal both 15 and 3, system is inconsistent ✓

Answer: -4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no solution" with the parallel lines concept. They might try to solve the system directly by elimination or substitution, get confused when variables cancel out leaving a false statement like \(\mathrm{15 = 3}\), but don't understand this means they need to work backwards to find what k value caused this.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for coefficient ratios but make sign errors when solving \(\mathrm{\frac{k}{-8} = \frac{1}{2}}\). They might forget the negative sign and calculate \(\mathrm{k = 4}\) instead of \(\mathrm{k = -4}\), or incorrectly cross multiply.

This may lead them to select k = 4 if that's an available option.

The Bottom Line:

This problem requires students to think backwards - instead of solving a system, they must determine what parameter value prevents a solution from existing. The key insight is recognizing that parallel lines (equal coefficient ratios) create the "no solution" condition.