Question:5x - 6/x = 13What is the value of the positive solution to the given equation?

1. TRANSLATE the problem information

• Given equation: \(5\mathrm{x} - \frac{6}{\mathrm{x}} = 13\)
• Find: positive solution value
• Note: \(\mathrm{x} \neq 0\) (since x appears in denominator)

2. INFER the solution strategy

• The fraction \(\frac{6}{\mathrm{x}}\) makes this challenging to solve directly
• Strategy: eliminate the fraction first by multiplying both sides by x
• This will create a quadratic equation we can solve

3. SIMPLIFY by eliminating the fraction

• Multiply both sides by x: \(\mathrm{x}(5\mathrm{x} - \frac{6}{\mathrm{x}}) = \mathrm{x}(13)\)
• Left side: \(\mathrm{x}(5\mathrm{x}) - \mathrm{x}(\frac{6}{\mathrm{x}}) = 5\mathrm{x}^2 - 6\)
• Right side: \(\mathrm{x}(13) = 13\mathrm{x}\)
• Result: \(5\mathrm{x}^2 - 6 = 13\mathrm{x}\)

4. SIMPLIFY to standard quadratic form

• Move all terms to left side: \(5\mathrm{x}^2 - 6 - 13\mathrm{x} = 0\)
• Rearrange: \(5\mathrm{x}^2 - 13\mathrm{x} - 6 = 0\)

5. INFER factoring approach and SIMPLIFY the quadratic

• For factoring, need two numbers that multiply to \(\mathrm{ac} = 5(-6) = -30\) and sum to \(\mathrm{b} = -13\)
• Those numbers are -15 and 2: \((-15)(2) = -30\) and \(-15 + 2 = -13\)
• Rewrite middle term: \(5\mathrm{x}^2 - 15\mathrm{x} + 2\mathrm{x} - 6 = 0\)
• Factor by grouping: \(5\mathrm{x}(\mathrm{x} - 3) + 2(\mathrm{x} - 3) = 0\)
• Factor completely: \((5\mathrm{x} + 2)(\mathrm{x} - 3) = 0\)

6. APPLY zero product property

• If \((5\mathrm{x} + 2)(\mathrm{x} - 3) = 0\), then either factor equals zero
• \(5\mathrm{x} + 2 = 0\) → \(\mathrm{x} = -\frac{2}{5}\)
• \(\mathrm{x} - 3 = 0\) → \(\mathrm{x} = 3\)

7. APPLY CONSTRAINTS to select final answer

• Both solutions are mathematically valid
• Question asks for the positive solution
• Since \(-\frac{2}{5}\) is negative and 3 is positive: \(\mathrm{x} = 3\)

Answer: C. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making algebraic errors when multiplying by x or rearranging to standard form

Students often make sign errors when moving terms around, leading to incorrect quadratics like \(5\mathrm{x}^2 + 13\mathrm{x} - 6 = 0\) instead of \(5\mathrm{x}^2 - 13\mathrm{x} - 6 = 0\). This changes the factoring completely and produces wrong solutions. They might end up with factors like \((5\mathrm{x} - 2)(\mathrm{x} + 3) = 0\), giving solutions \(\mathrm{x} = \frac{2}{5}\) and \(\mathrm{x} = -3\), leading them to select Choice A (2/5) as the positive answer.

Second Most Common Error:

Poor INFER reasoning: Attempting to solve the rational equation without eliminating the fraction first

Some students try to work with \(5\mathrm{x} - \frac{6}{\mathrm{x}} = 13\) directly, perhaps trying to isolate x in steps without a clear strategy. They might manipulate it incorrectly, getting confused about how to handle the \(\frac{6}{\mathrm{x}}\) term, and eventually abandon systematic solution. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires systematic algebraic manipulation through multiple steps. The key insight is recognizing that multiplying by x transforms a challenging rational equation into a familiar quadratic equation that can be factored and solved using standard techniques.