66x = 66x How many solutions does the given equation have?...

1. INFER the nature of the equation

• Given: \(\mathrm{66x = 66x}\)
• Key observation: Both sides of the equation are completely identical
• This means we have the exact same expression on both sides

2. INFER what this means for solutions

• When both sides of an equation are identical, the equation becomes an identity
• An identity is true no matter what value we substitute for the variable
• Let's test this understanding with a few values:
  • If x = 0: \(\mathrm{66(0) = 66(0)}\) → \(\mathrm{0 = 0}\) ✓
  • If x = 5: \(\mathrm{66(5) = 66(5)}\) → \(\mathrm{330 = 330}\) ✓
  • If x = -1: \(\mathrm{66(-1) = 66(-1)}\) → \(\mathrm{-66 = -66}\) ✓

3. INFER the final conclusion

• Since the equation is true for every possible value of x
• This means x can be any real number
• Therefore, the equation has infinitely many solutions

Answer: C. Infinitely many

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see "\(\mathrm{66x = 66x}\)" and think "this looks like it should have exactly one solution" because many equations they've solved before had exactly one solution.

They might reason: "Most equations I solve have one answer, so this must too." They fail to recognize that identical sides create a special case - an identity that's always true.

This may lead them to select Choice A (Exactly one).

Second Most Common Error:

Conceptual confusion about equation types: Students might think that because the equation "looks weird" or "too simple," something must be wrong, leading them to conclude it has no solutions.

They might reason: "This equation doesn't make sense - it's the same thing on both sides, so maybe there's no solution."

This may lead them to select Choice D (Zero).

The Bottom Line:

This problem tests whether students understand that equations can be identities. The key insight is recognizing that when both sides are identical, you've found an equation that's true for ALL values, not just one specific value.