\(\mathrm{a(3 - x) - b = -1 - 2x}\)In the equation above, a and b are constants. If the equation...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{a(3 - x) - b = -1 - 2x}\)
• We need values of constants a and b that make this equation have infinitely many solutions

2. INFER what "infinitely many solutions" means

• An equation has infinitely many solutions when it's an identity
• This happens when both sides are exactly the same after simplification
• The coefficients of like terms must be equal on both sides

3. SIMPLIFY by expanding the left side

• Distribute the a: \(\mathrm{a(3 - x) - b = 3a - ax - b}\)
• Our equation becomes: \(\mathrm{3a - ax - b = -1 - 2x}\)

4. SIMPLIFY by rearranging to group like terms

• Move x-terms and constants together: \(\mathrm{-ax + 3a - b = -2x - 1}\)
• Now we can compare coefficients directly

5. INFER the required relationships

• For the equation to be an identity:
  • Coefficient of x on left = Coefficient of x on right
  • Constant term on left = Constant term on right

6. SIMPLIFY to find the values

• Coefficients of x: \(\mathrm{-a = -2}\), so \(\mathrm{a = 2}\)
• Constant terms: \(\mathrm{3a - b = -1}\)
• Substitute \(\mathrm{a = 2}\): \(\mathrm{3(2) - b = -1}\)
• Solve: \(\mathrm{6 - b = -1}\), so \(\mathrm{b = 7}\)

Answer: B. a = 2 and b = 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize what "infinitely many solutions" means mathematically. They might try to solve for specific values of x instead of understanding that the equation must be an identity. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when distributing or rearranging terms. For example, they might get the wrong sign for the coefficient of x (writing \(\mathrm{+a = -2}\) instead of \(\mathrm{-a = -2}\)) or make errors in the constant terms. This may lead them to select Choice A (a = 2, b = 1) or Choice C (a = -2, b = 5).

The Bottom Line:

This problem tests whether students understand the deeper meaning of "infinitely many solutions" - it's not about finding x-values, but about making the equation an identity through proper coefficient matching.