-{49x = -98x} How many solutions does the given equation have?...

1. TRANSLATE the problem information

• Given equation: \(-49\mathrm{x} = -98\mathrm{x}\)
• Need to find: How many solutions this equation has

2. SIMPLIFY by collecting like terms

• Add 98x to both sides to get all x terms on one side:
  \(-49\mathrm{x} + 98\mathrm{x} = -98\mathrm{x} + 98\mathrm{x}\)
  \(49\mathrm{x} = 0\)

3. SIMPLIFY further to solve for x

• Divide both sides by 49:
  \(\mathrm{x} = 0\)

4. INFER what this result means

• Since we found exactly one value (\(\mathrm{x} = 0\)) that makes the equation true, the equation has exactly one solution

Answer: B. Exactly one

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when moving terms from one side to the other. They might incorrectly think "add 98x" means the left side becomes \(-49\mathrm{x} - 98\mathrm{x} = -147\mathrm{x}\), leading to \(-147\mathrm{x} = 0\), so \(\mathrm{x} = 0\). While they still get \(\mathrm{x} = 0\), the error in process could cause confusion in more complex problems.

Second Most Common Error:

Conceptual confusion about solution types: Students see variables on both sides and mistakenly think this means "infinitely many solutions" like in identity equations (such as \(2\mathrm{x} = 2\mathrm{x}\)). They don't recognize that this particular equation simplifies to a specific value.

This may lead them to select Choice D (Infinitely many)

The Bottom Line:

The key insight is recognizing that even though both sides contain x terms, this equation reduces to a simple statement (\(49\mathrm{x} = 0\)) that has exactly one solution, not infinitely many like true identities.