The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of x?3x - 6 =...

1. INFER what the problem is actually asking

• Given information:
  • \(\mathrm{3x - 6 = 21}\) (linear equation)
  • \(\mathrm{y = 2x^2 - 5x + 7}\) (quadratic equation)
  • Need to find: value of x
• Key insight: Since we only need x, and the first equation contains only x, we can solve it directly without touching the quadratic equation.

2. SIMPLIFY the linear equation to find x

• Starting with: \(\mathrm{3x - 6 = 21}\)
• Add 6 to both sides: \(\mathrm{3x = 27}\)
• Divide both sides by 3: \(\mathrm{x = 9}\)

3. Verify the solution (optional but recommended)

• Check in first equation: \(\mathrm{3(9) - 6 = 27 - 6 = 21}\) ✓
• Check in second equation: \(\mathrm{y = 2(9)^2 - 5(9) + 7 = 2(81) - 45 + 7 = 124}\) ✓

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students overthink the "system" aspect and believe they must solve both equations simultaneously using substitution or elimination methods.

They might try to substitute the quadratic expression for y into the linear equation, not realizing that the linear equation doesn't contain y at all. This creates unnecessary complexity and confusion about how to proceed, often leading them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when solving the simple linear equation.

Common mistakes include: \(\mathrm{3x - 6 = 21 \rightarrow 3x = 21 - 6 = 15 \rightarrow x = 5}\), or forgetting to divide by 3 and thinking \(\mathrm{x = 27}\). This may lead them to select Choice A (3) or Choice D (27).

The Bottom Line:

This problem tests whether students can recognize that a "system" doesn't always require complex solving methods - sometimes one equation gives you exactly what you need. The key is identifying which equation contains your target variable and solving it directly.