The system of equations below contains the parameter n.5x + ny = 20nx + 5y = 12If there are multiple...

1. TRANSLATE the problem information

• Given system:
  • \(5\mathrm{x} + \mathrm{n}\mathrm{y} = 20\)
  • \(\mathrm{n}\mathrm{x} + 5\mathrm{y} = 12\)
• Find: positive value of n where system has no solution

2. INFER when a system has no solution

• A system has no solution when the equations represent parallel but different lines
• This happens when coefficient ratios are equal but constant ratios are different
• For our system: coefficient ratio = \(\frac{5}{\mathrm{n}} = \frac{\mathrm{n}}{5}\)

3. SIMPLIFY to find possible n values

• Set up the proportion: \(\frac{5}{\mathrm{n}} = \frac{\mathrm{n}}{5}\)
• Cross multiply: \(5 \times 5 = \mathrm{n} \times \mathrm{n}\)
• Simplify: \(25 = \mathrm{n}^2\)
• Take square root: \(\mathrm{n} = \pm5\)

4. CONSIDER ALL CASES by verifying both solutions

For n = 5:

• System becomes: \(5\mathrm{x} + 5\mathrm{y} = 20\) and \(5\mathrm{x} + 5\mathrm{y} = 12\)
• Dividing by 5: \(\mathrm{x} + \mathrm{y} = 4\) and \(\mathrm{x} + \mathrm{y} = 2.4\)
• Since \(4 \neq 2.4\), these equations are inconsistent → no solution ✓

For n = -5:

• System becomes: \(5\mathrm{x} - 5\mathrm{y} = 20\) and \(-5\mathrm{x} + 5\mathrm{y} = 12\)
• First equation: \(\mathrm{x} - \mathrm{y} = 4\)
• Second equation: \(-\mathrm{x} + \mathrm{y} = 2.4\) → \(\mathrm{x} - \mathrm{y} = -2.4\)
• Since \(4 \neq -2.4\), these equations are inconsistent → no solution ✓

5. APPLY CONSTRAINTS to select final answer

• Both \(\mathrm{n} = 5\) and \(\mathrm{n} = -5\) create systems with no solution
• Problem asks for the positive value
• Therefore: \(\mathrm{n} = 5\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the condition for no solution, instead trying to solve the system directly or setting the determinant equal to zero without understanding what this means.

They might attempt to solve for x and y algebraically, get confused when the system seems unsolvable, and then guess randomly. This leads to confusion and guessing.

Second Most Common Error:

Incomplete CONSIDER ALL CASES execution: Students correctly find \(\mathrm{n}^2 = 25\) but only consider \(\mathrm{n} = 5\), missing that \(\mathrm{n} = -5\) also works. They might think there's only one answer since the problem asks for 'the positive value.'

However, the problem states there are 'multiple values of n' that work, so they need to find both values first, then select the positive one. Missing this verification step may lead them to doubt their work and select an incorrect answer.

The Bottom Line:

This problem tests understanding of when linear systems fail to have solutions. The key insight is recognizing that 'no solution' means parallel but non-identical lines, which translates to a specific relationship between coefficients that students must identify and solve.