One of the two equations in a linear system is 2x + 6y = 10. The system has no solution....

1. INFER what 'no solution' means for linear systems

• Given information:
  • One equation: \(\mathrm{2x + 6y = 10}\)
  • The system has no solution

• What this tells us: For no solution, we need parallel lines that never intersect. This happens when equations have the same coefficient ratios but are not equivalent.

2. SIMPLIFY by finding the coefficient ratio pattern

• From \(\mathrm{2x + 6y = 10}\), the ratio of x-coefficient to y-coefficient is:
  \(\mathrm{2:6 = 1:3}\)

• For no solution, the other equation must also have ratio \(\mathrm{1:3}\) but be non-equivalent

3. INFER which choice fits the criteria

• Check each option for the \(\mathrm{1:3}\) ratio:

Choice A: \(\mathrm{x + 3y = 5}\)

• Ratio: \(\mathrm{1:3}\) ✓
• Multiply by 2: \(\mathrm{2x + 6y = 10}\) (identical to given equation)
• This gives infinite solutions, not no solution ✗

Choice B: \(\mathrm{x + 3y = -20}\)

• Ratio: \(\mathrm{1:3}\) ✓
• Multiply by 2: \(\mathrm{2x + 6y = -40}\) (different from given equation)
• Same ratios, different equations = no solution ✓

Choice C: \(\mathrm{6x - 2y = 0}\)

• Ratio: \(\mathrm{6:(-2) = -3:1}\) (different from \(\mathrm{1:3}\)) ✗

Choice D: \(\mathrm{6x + 2y = 10}\)

• Ratio: \(\mathrm{6:2 = 3:1}\) (different from \(\mathrm{1:3}\)) ✗

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse 'no solution' with 'infinite solutions'

They correctly identify that Choice A (\(\mathrm{x + 3y = 5}\)) has the same coefficient ratio as the given equation, but fail to recognize that multiplying by 2 gives exactly the same equation (\(\mathrm{2x + 6y = 10}\)). Same equations mean infinite solutions, not no solution.

This may lead them to select Choice A (\(\mathrm{x + 3y = 5}\))

Second Most Common Error:

Incomplete SIMPLIFY execution: Students focus only on making coefficients 'look different'

They might pick Choice C or D because the coefficients appear more different from the original equation, without checking whether the coefficient ratios create parallel lines.

This may lead them to select Choice C (\(\mathrm{6x - 2y = 0}\)) or Choice D (\(\mathrm{6x + 2y = 10}\))

The Bottom Line:

No solution requires a delicate balance: equations must be similar enough to be parallel (same coefficient ratios) but different enough to never intersect (non-equivalent equations). Students often miss this subtle distinction.