What is the solution to the given equation? \(\frac{(\mathrm{x}+9)(\mathrm{x}-9)}{(\mathrm{x}+9)} = 7\)...

1. SIMPLIFY the rational expression

• Given equation: \(\frac{(x+9)(x-9)}{(x+9)} = 7\)
• Key insight: Both numerator and denominator contain the factor \((x+9)\)
• Cancel the common factors: \(\frac{(x+9)(x-9)}{(x+9)} = x-9\)
• Simplified equation: \(x - 9 = 7\)

2. SIMPLIFY the linear equation

• Solve for x by adding 9 to both sides:
  \(x - 9 + 9 = 7 + 9\)
  \(x = 16\)

3. Verify the solution

• Substitute \(x = 16\) back into original equation:
  \(\frac{(16+9)(16-9)}{(16+9)} = \frac{(25)(7)}{(25)} = \frac{175}{25} = 7\) ✓

Answer: C. 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students don't recognize they can cancel the common \((x+9)\) factors and instead try to solve the equation as written.

They might attempt to multiply out \((x+9)(x-9) = x^2 - 81\), leading to the complex equation \(\frac{x^2 - 81}{x+9} = 7\), then cross-multiply to get \(x^2 - 81 = 7(x+9) = 7x + 63\), resulting in \(x^2 - 7x - 144 = 0\). This unnecessarily complicated quadratic approach often leads to calculation errors and confusion.

This may lead them to select incorrect answers or abandon the systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the cancellation opportunity but make algebraic errors in the process.

They might incorrectly cancel terms (perhaps thinking the entire numerator cancels with the denominator) or make sign errors when solving \(x - 9 = 7\) (getting \(x = -2\) instead of \(x = 16\)).

This may lead them to select Choice A (7) if they make multiple errors that coincidentally produce this value.

The Bottom Line:

The key to this problem is recognizing that rational expressions can often be simplified dramatically by canceling common factors. Students who miss this simplification step make the problem much harder than it needs to be.