What is the solution to the given system of equations?y = 12x - 20y = 28

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = 12x - 20}\)
  • \(\mathrm{y = 28}\)
• We need to find the values of x and y that satisfy both equations simultaneously

2. INFER the solution strategy

• Since both equations give us expressions for y, we can use substitution
• Key insight: If \(\mathrm{y = 28}\) and \(\mathrm{y = 12x - 20}\), then \(\mathrm{28 = 12x - 20}\)

3. SIMPLIFY the resulting equation

• Start with: \(\mathrm{28 = 12x - 20}\)
• Add 20 to both sides: \(\mathrm{28 + 20 = 12x}\)
• This gives us: \(\mathrm{48 = 12x}\)
• Divide both sides by 12: \(\mathrm{x = 4}\)

4. Find the y-coordinate

• From the second equation: \(\mathrm{y = 28}\)
• Our solution point is \(\mathrm{(4, 28)}\)

5. APPLY CONSTRAINTS to verify format

• Check: Does \(\mathrm{(4, 28)}\) satisfy both equations?
  • First equation: \(\mathrm{y = 12(4) - 20}\)
    \(\mathrm{= 48 - 20}\)
    \(\mathrm{= 28}\) ✓
  • Second equation: \(\mathrm{y = 28}\) ✓
• Solution must be written as \(\mathrm{(x, y) = (4, 28)}\)

Answer: A. \(\mathrm{(4, 28)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors when solving \(\mathrm{28 = 12x - 20}\)

Students might incorrectly add/subtract when isolating x, leading to wrong values like \(\mathrm{x = 20}\) or other incorrect calculations. This may lead them to select Choice B \(\mathrm{(20, 28)}\) or Choice D \(\mathrm{(28, 20)}\).

Second Most Common Error:

Poor understanding of ordered pair notation: Getting confused about which value is x and which is y

Students correctly find \(\mathrm{x = 4}\) and \(\mathrm{y = 28}\) but write the answer as \(\mathrm{(28, 4)}\) instead of \(\mathrm{(4, 28)}\). This leads them to select Choice C \(\mathrm{(28, 4)}\).

The Bottom Line:

This problem tests whether students can efficiently use substitution for systems where one equation already isolates a variable, and whether they understand ordered pair notation. The key insight is recognizing the direct substitution opportunity rather than getting bogged down in more complex elimination methods.