What is the solution to the given equation? (-54)/w = 6...

1. INFER the solution strategy

Given equation: \(\frac{-54}{\mathrm{w}} = 6\)

• Strategy: Since w appears in the denominator of a fraction, we need to eliminate the fraction first by multiplying both sides by w, then isolate w.
• Important note: Since w is in the denominator, \(\mathrm{w} ≠ 0\).

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by w:
  \(\frac{-54}{\mathrm{w}} × \mathrm{w} = 6 × \mathrm{w}\)
  \(-54 = 6\mathrm{w}\)
• This eliminates the fraction and gives us a simpler linear equation.

3. SIMPLIFY by isolating the variable

• Divide both sides by 6:
  \(\frac{-54}{6} = \frac{6\mathrm{w}}{6}\)
  \(-9 = \mathrm{w}\)
• Therefore: \(\mathrm{w} = -9\)

4. Verify the solution

• Check: \(\frac{-54}{-9} = \frac{54}{9} = 6\) ✓
• Also confirm \(\mathrm{w} ≠ 0\): Since \(-9 ≠ 0\), our solution is valid.

Answer: -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when working with negative numbers

Students often struggle with the arithmetic involving negative numbers, particularly when dividing -54 by 6. They might incorrectly calculate \(\frac{-54}{6} = 9\) instead of \(\frac{-54}{6} = -9\), or make errors when substituting back to check their work.

This leads them to an incorrect answer of \(\mathrm{w} = 9\), which doesn't match any typical wrong answer pattern but causes confusion during verification.

Second Most Common Error:

Poor INFER reasoning: Attempting to solve by cross-multiplication incorrectly

Some students might try to rewrite \(\frac{-54}{\mathrm{w}} = 6\) as \(-54 = 6\mathrm{w}\) immediately without proper justification, or attempt other algebraic manipulations without a clear strategy. This can lead to computational errors or getting stuck mid-solution.

This causes them to abandon systematic solution and resort to guessing.

The Bottom Line:

This problem tests fundamental equation-solving skills with rational expressions. Success requires both strategic thinking (recognizing how to eliminate fractions) and careful arithmetic with negative numbers.