y = 6x + 3 One of the two equations in a system of linear equations is given. The system...

1. INFER what "infinitely many solutions" means

• Given information:
  • First equation: \(\mathrm{y = 6x + 3}\)
  • The system has infinitely many solutions
• What this tells us: For a system to have infinitely many solutions, both equations must represent the same line (they must be equivalent)

2. INFER the strategy

• Since equivalent equations are multiples of each other, I need to check which answer choice becomes identical to \(\mathrm{y = 6x + 3}\) when simplified
• I'll SIMPLIFY each option to see which one reduces to the original equation

3. SIMPLIFY each answer choice

Choice A: \(\mathrm{y = 2(6x) + 3}\)

• Distribute: \(\mathrm{y = 12x + 3}\)
• This has a different slope (12 vs 6), so it's NOT equivalent

Choice B: \(\mathrm{y = 2(6x + 3)}\)

• Distribute: \(\mathrm{y = 12x + 6}\)
• This has a different slope AND y-intercept, so it's NOT equivalent

Choice C: \(\mathrm{2(y) = 2(6x) + 3}\)

• Simplify: \(\mathrm{2y = 12x + 3}\)
• Divide by 2: \(\mathrm{y = 6x + \frac{3}{2}}\)
• This has a different y-intercept (3/2 vs 3), so it's NOT equivalent

Choice D: \(\mathrm{2(y) = 2(6x + 3)}\)

• Distribute the right side: \(\mathrm{2y = 2(6x) + 2(3) = 12x + 6}\)
• Divide both sides by 2: \(\mathrm{y = 6x + 3}\)
• This is identical to our original equation!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "infinitely many solutions" to "equivalent equations." Instead, they might think any two different-looking equations could work, or they might not understand what infinitely many solutions means in the context of linear systems.

This leads to confusion about the strategy, causing them to guess randomly among the choices rather than systematically checking for equivalence.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand they need equivalent equations but make algebraic errors when distributing or simplifying. For example, in Choice C, they might incorrectly simplify \(\mathrm{2y = 12x + 3}\) to \(\mathrm{y = 6x + 3}\) (forgetting to divide the constant term by 2), leading them to incorrectly select Choice C.

The Bottom Line:

This problem tests both conceptual understanding of what "infinitely many solutions" means AND the algebraic skills to verify equivalence. Students need to connect the abstract concept to the concrete algebraic manipulation.