\(\mathrm{(b - 2)x = 8}\) In the given equation, b is a constant. If the equation has no solution, what...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{(b - 2)x = 8}\)
• Need to find: value of b that makes this equation have no solution
• Answer choices: 2, 4, 6, or 10

2. INFER when linear equations have no solution

• A linear equation has no solution when we get a contradiction
• This happens when the coefficient of x equals 0 but the constant term doesn't equal 0
• Strategy: Find when the coefficient (b - 2) equals zero

3. SIMPLIFY to find the critical value

• Set the coefficient equal to zero: \(\mathrm{b - 2 = 0}\)
• Solve: \(\mathrm{b = 2}\)

4. INFER what happens when b = 2

• Substitute \(\mathrm{b = 2}\) into original equation: \(\mathrm{(2 - 2)x = 8}\)
• This gives us: \(\mathrm{0x = 8}\)
• Since 0x always equals 0, we have: \(\mathrm{0 = 8}\)
• This is impossible! No value of x can make this true.

5. Verify other choices give exactly one solution

• \(\mathrm{b = 4}\): \(\mathrm{(4-2)x = 8}\) → \(\mathrm{2x = 8}\) → \(\mathrm{x = 4}\) ✓
• \(\mathrm{b = 6}\): \(\mathrm{(6-2)x = 8}\) → \(\mathrm{4x = 8}\) → \(\mathrm{x = 2}\) ✓
• \(\mathrm{b = 10}\): \(\mathrm{(10-2)x = 8}\) → \(\mathrm{8x = 8}\) → \(\mathrm{x = 1}\) ✓

Answer: A. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between "no solution" and getting a contradiction like \(\mathrm{0 = 8}\). Instead, they might think "no solution" means \(\mathrm{x = 0}\), so they try to solve \(\mathrm{(b-2)(0) = 8}\), which gives \(\mathrm{0 = 8}\), but then they incorrectly think any value of b works since they get \(\mathrm{0 = 8}\) regardless.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about solving equations: Students might think that having "no solution" means the equation can't be solved for x, so they look for values of b that make x very large or very small. They might substitute each answer choice and calculate x, then pick the largest value thinking that represents "no solution."

This may lead them to select Choice D (10) since it gives the smallest value \(\mathrm{x = 1}\), or they might randomly guess.

The Bottom Line:

This problem requires recognizing that "no solution" in linear equations specifically means getting a mathematical impossibility (like \(\mathrm{0 = 8}\)), not just a difficult calculation. The key insight is understanding what makes an equation unsolvable rather than just hard to solve.