In the xy-plane, the graph of y = x + 3 intersects the graph of y = 2x - 6...

1. TRANSLATE the problem information

• Given information:
  • Two linear equations: \(\mathrm{y = x + 3}\) and \(\mathrm{y = 2x - 6}\)
  • These graphs intersect at point \(\mathrm{(a, b)}\)
  • We need to find the value of a

2. INFER the mathematical approach

• Key insight: At the intersection point, both equations must be satisfied simultaneously
• This means both expressions for y are equal at the intersection
• Strategy: Set the right sides of the equations equal to each other

3. Set up the equation

• Since both equal y: \(\mathrm{x + 3 = 2x - 6}\)

4. SIMPLIFY to solve for x

• \(\mathrm{x + 3 = 2x - 6}\)
• Add 6 to both sides: \(\mathrm{x + 9 = 2x}\)
• Subtract x from both sides: \(\mathrm{9 = x}\)
• Therefore: \(\mathrm{a = 9}\)

5. Verify the answer (optional but recommended)

• Substitute \(\mathrm{x = 9}\) into both original equations:
  • \(\mathrm{y = 9 + 3 = 12}\)
  • \(\mathrm{y = 2(9) - 6 = 18 - 6 = 12}\) ✓
• Both give the same y-value, confirming our intersection point is \(\mathrm{(9, 12)}\)

Answer: C. 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that intersection means setting the equations equal to each other. Some students might try to solve each equation individually or get confused about what "intersection" means mathematically. Without this key insight, they may attempt various unsuccessful approaches or simply guess.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when solving \(\mathrm{x + 3 = 2x - 6}\). Common mistakes include:

• Incorrectly moving terms: getting \(\mathrm{x + 3 + 6 = 2x}\) instead of \(\mathrm{x + 9 = 2x}\)
• Sign errors when subtracting or adding terms
• Mixing up which variable they're solving for

This may lead them to select Choice A (3) or Choice B (6) depending on the specific algebraic mistake.

The Bottom Line:

This problem tests whether students understand that intersection points satisfy both equations simultaneously and can translate this understanding into the algebraic step of setting expressions equal to each other.