Question:y = 4x - 212x - 3y = 6At how many points do the graphs of these two equations intersect...

1. TRANSLATE the problem information

• Given equations:
  • \(\mathrm{y = 4x - 2}\)
  • \(\mathrm{12x - 3y = 6}\)
• Question asks: How many intersection points do these graphs have?

2. INFER the approach

• To find intersection points, we need to solve the system of equations
• We can use substitution since the first equation already gives us y in terms of x
• The number of solutions tells us about intersection points

3. SIMPLIFY using substitution

• Substitute \(\mathrm{y = 4x - 2}\) into the second equation:
  \(\mathrm{12x - 3(4x - 2) = 6}\)
• Distribute the -3:
  \(\mathrm{12x - 12x + 6 = 6}\)
• Combine like terms:
  \(\mathrm{6 = 6}\)

4. INFER what this result means

• Getting "\(\mathrm{6 = 6}\)" is an identity—it's always true
• This means the equations are dependent (they're actually the same line)
• When two equations represent the same line, they intersect at every point on that line

5. Verify by converting to slope-intercept form

• SIMPLIFY the second equation:
  \(\mathrm{12x - 3y = 6}\)
  \(\mathrm{-3y = -12x + 6}\)
  \(\mathrm{y = 4x - 2}\)
• Both equations are identical: \(\mathrm{y = 4x - 2}\)

Answer: D) Infinitely many

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see "\(\mathrm{6 = 6}\)" and think it means there's no solution, confusing dependent systems with inconsistent systems.

When you get an equation like "\(\mathrm{6 = 6}\)" (always true), it means the original equations are dependent—they're the same line. But students often think "something weird happened, so there must be no intersection." An inconsistent system would give something like "\(\mathrm{6 = 0}\)" (never true), which means parallel lines with no intersection.

This may lead them to select Choice A (Zero).

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors during substitution, particularly when distributing negative signs.

For example, they might incorrectly calculate:
\(\mathrm{12x - 3(4x - 2) = 6}\) as \(\mathrm{12x - 12x - 6 = 6}\) (forgetting to distribute the negative to both terms), leading to \(\mathrm{-6 = 6}\), which would suggest no solution.

This leads to confusion and may cause them to select Choice A (Zero).

The Bottom Line:

The key insight is recognizing what different algebraic outcomes mean geometrically. An identity means the same line (infinitely many intersections), while a contradiction means parallel lines (zero intersections).