If 28/(7y - 35) is equivalent to 4/(y - b) for all values of y for which both expressions are...

1. TRANSLATE the problem information

• Given: \(\frac{28}{7y - 35}\) is equivalent to \(\frac{4}{y - b}\) for all valid values of y
• Find: the constant b

2. SIMPLIFY the first expression

• Factor the denominator: \(7y - 35 = 7(y - 5)\)
• Rewrite: \(\frac{28}{7y - 35} = \frac{28}{7(y - 5)}\)
• Simplify the fraction: \(\frac{28}{7(y - 5)} = (28 \div 7) \times \frac{1}{y - 5} = \frac{4}{y - 5}\)

3. INFER the relationship for equivalence

• Now we have: \(\frac{4}{y - 5} = \frac{4}{y - b}\)
• For rational expressions to be equivalent for ALL values of y, their simplified forms must be identical
• This means the denominators must match exactly

4. SIMPLIFY to find b

• Set denominators equal: \(y - 5 = y - b\)
• Subtract y from both sides: \(-5 = -b\)
• Therefore: \(b = 5\)

5. SIMPLIFY verification using cross-multiplication

• Check: \(\frac{28}{7y - 35} = \frac{4}{y - 5}\) when \(b = 5\)
• Cross-multiply: \(28(y - 5) = 4(7y - 35)\)
• Expand: \(28y - 140 = 28y - 140\) ✓

Answer: B (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students fail to properly factor \(7y - 35\) or make arithmetic errors when simplifying \(\frac{28}{7}\).

For example, they might incorrectly factor as \(7y - 35 = 7(y - 35)\) or calculate \(\frac{28}{7} = 3\). This leads to wrong simplified forms like \(\frac{4}{y - 35}\) or \(\frac{3}{y - 5}\), causing them to find incorrect values of b.

This may lead them to select Choice E (35) if they use the incorrect factor, or get confused and guess.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that equivalent expressions must have identical simplified denominators.

Instead, they might try to match the original expressions directly, setting \(7y - 35 = y - b\), which gives \(6y - 35 = b\). This approach ignores the fundamental requirement for rational expression equivalence.

This causes them to get stuck trying to solve for a specific value of b when the relationship depends on y, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students understand that rational expression equivalence requires matching simplified forms, not just algebraic manipulation of the original expressions. The key insight is recognizing when and how to simplify before comparing.