If \(\frac{5}{\mathrm{x} - 3} = \frac{4}{\mathrm{x} - 3} + \frac{29}{(\mathrm{x} - 3)^2}\), what is the value of x - 3?

1. INFER the most efficient approach

• Looking at the equation \(\frac{5}{x - 3} = \frac{4}{x - 3} + \frac{29}{(x - 3)^2}\)
• Key insight: The expression \((x-3)\) appears multiple times
• Strategy: Use substitution to simplify before solving

2. TRANSLATE using substitution

• Let \(y = x - 3\)
• The equation becomes: \(\frac{5}{y} = \frac{4}{y} + \frac{29}{y^2}\)
• This is much cleaner to work with!

3. SIMPLIFY by clearing denominators

• Multiply both sides by \(y^2\):
  • Left side: \(y^2 \cdot \frac{5}{y} = 5y\)
  • Right side: \(y^2 \cdot \frac{4}{y} + y^2 \cdot \frac{29}{y^2} = 4y + 29\)
• Result: \(5y = 4y + 29\)

4. SIMPLIFY to solve for y

• Subtract \(4y\) from both sides: \(5y - 4y = 29\)
• Therefore: \(y = 29\)

5. TRANSLATE back to find the answer

• Since \(y = x - 3\) and \(y = 29\)
• We have \(x - 3 = 29\)

Answer: B (29)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the substitution opportunity and trying to work directly with the \((x-3)\) terms.

Students might attempt to find a common denominator of \((x-3)^2\) for all terms, leading to:
\(\frac{5(x-3)}{(x-3)^2} = \frac{4(x-3)}{(x-3)^2} + \frac{29}{(x-3)^2}\)

While this approach can work, it's much more prone to algebraic errors and doesn't simplify the problem structure. This leads to confusion and often causes students to make computational mistakes or abandon the systematic approach entirely.

Second Most Common Error:

Poor SIMPLIFY execution: Making errors when multiplying by \(y^2\) to clear denominators.

A common mistake is incorrectly distributing \(y^2\), such as getting \(5y^2 = 4y^2 + 29\) instead of \(5y = 4y + 29\). This leads to a quadratic equation instead of the simple linear equation, and students might select Choice C (32) if they solve the incorrect quadratic.

The Bottom Line:

The key to this problem is recognizing that substitution transforms a complex-looking rational equation into a straightforward linear equation. Students who miss this strategic insight often get bogged down in unnecessary algebraic complexity.