The expression 21/(35 - 7x) is equivalent to the expression 3/(b - x), where b is a constant and x...

1. TRANSLATE the problem information

• Given information:
  • Expression 1: \(\frac{21}{35 - 7x}\)
  • Expression 2: \(\frac{3}{b - x}\)
  • These expressions are equivalent
  • Need to find the value of b

2. INFER the approach

• Since the expressions are equivalent, I can either:
  • Simplify the first expression to match the second form, OR
  • Set them equal and cross-multiply
• I'll use the factoring approach first since it's more direct

3. SIMPLIFY by factoring the denominator

• Look at \(35 - 7x\): both terms have a common factor of 7
• Factor out 7: \(35 - 7x = 7(5 - x)\)
• Substitute: \(\frac{21}{35 - 7x} = \frac{21}{7(5 - x)}\)

4. SIMPLIFY the fraction

• Divide numerator and denominator: \(21 ÷ 7 = 3\)
• Result: \(\frac{3}{5 - x}\)

5. INFER the final comparison

• Now I have: \(\frac{3}{5 - x} = \frac{3}{b - x}\)
• Since the numerators are identical, the denominators must be equal
• Therefore: \(b - x = 5 - x\)
• This means: \(b = 5\)

Answer: C (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with factoring the denominator correctly, particularly missing that both 35 and 7x share a common factor of 7. They might try to factor as \(7(5 - x)\) but write it as \(7(5 + x)\) due to sign confusion, or they might not recognize the factoring opportunity at all.

This leads to confusion and inability to simplify the expression, causing them to get stuck and guess.

Second Most Common Error:

Poor INFER reasoning: Students attempt cross-multiplication but make sign errors when distributing. For example, when expanding \(3(35 - 7x)\), they might write \(105 + 21x\) instead of \(105 - 21x\), leading to an incorrect equation and wrong value for b.

This may lead them to select Choice A (-5) or Choice B (7) depending on the specific algebraic error made.

The Bottom Line:

This problem tests whether students can recognize equivalent forms of rational expressions. The key insight is that factoring creates the path to the solution - without seeing that 7 is a common factor in the denominator, students get stuck trying other approaches that are much more complex.