y = 12 - 18/x The given equation relates the numbers x and y, where x neq 0 and y...

1. INFER the solution strategy

• Goal: Isolate x from the equation \(\mathrm{y = 12 - \frac{18}{x}}\)
• Key insight: Since x is in the denominator, we need to eliminate the fraction first
• Strategy: Use properties of equality to move terms and clear the denominator

2. SIMPLIFY by moving the fraction term

• Starting equation: \(\mathrm{y = 12 - \frac{18}{x}}\)
• Add \(\mathrm{\frac{18}{x}}\) to both sides: \(\mathrm{y + \frac{18}{x} = 12}\)
• Subtract y from both sides: \(\mathrm{\frac{18}{x} = 12 - y}\)

3. SIMPLIFY by clearing the denominator

• Current equation: \(\mathrm{\frac{18}{x} = 12 - y}\)
• Multiply both sides by x: \(\mathrm{18 = x(12 - y)}\)
• This eliminates the fraction completely

4. SIMPLIFY to isolate x

• Current equation: \(\mathrm{18 = x(12 - y)}\)
• Divide both sides by (12 - y): \(\mathrm{x = \frac{18}{12 - y}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when rearranging terms, particularly getting confused about \(\mathrm{y - 12}\) vs \(\mathrm{12 - y}\).

When subtracting y from both sides of \(\mathrm{y + \frac{18}{x} = 12}\), some students incorrectly write \(\mathrm{\frac{18}{x} = y - 12}\) instead of \(\mathrm{\frac{18}{x} = 12 - y}\). Following through with correct algebra from this point gives \(\mathrm{x = \frac{18}{y - 12}}\).

This leads them to select Choice D (\(\mathrm{x = \frac{18}{y - 12}}\))

Second Most Common Error:

Poor SIMPLIFY reasoning: Students attempt to "flip" the fraction relationship incorrectly.

From \(\mathrm{\frac{18}{x} = 12 - y}\), some students think they can simply flip both sides to get \(\mathrm{\frac{x}{18} = 12 - y}\), then multiply by 18 to get \(\mathrm{x = \frac{12 - y}{18}}\). This shows confusion about reciprocal relationships in equations.

This may lead them to select Choice A (\(\mathrm{x = \frac{12 - y}{18}}\))

The Bottom Line:

This problem tests careful algebraic manipulation with attention to signs and proper fraction handling. The key is systematic step-by-step work rather than trying to take shortcuts that can introduce errors.