The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of y?3x = 12-{3x +...

1. TRANSLATE the problem information

• Given system:
  • Equation 1: \(\mathrm{3x = 12}\)
  • Equation 2: \(\mathrm{-3x + y = -6}\)
• Need to find: the value of y

2. INFER the solution strategy

• Notice the x-coefficients: \(\mathrm{+3x}\) in the first equation and \(\mathrm{-3x}\) in the second
• These are opposites, which means elimination will work perfectly
• When we add the equations, the x-terms will cancel out, leaving just y

3. SIMPLIFY by adding the equations

• Add left sides: \(\mathrm{3x + (-3x + y) = 0 + y = y}\)
• Add right sides: \(\mathrm{12 + (-6) = 6}\)
• Result: \(\mathrm{y = 6}\)

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elimination opportunity and instead try substitution, making the problem unnecessarily complicated.

They might solve \(\mathrm{3x = 12}\) to get \(\mathrm{x = 4}\), then substitute into the second equation: \(\mathrm{-3(4) + y = -6}\). While this works, it involves more steps and creates more opportunities for arithmetic errors. This approach is still correct but less efficient.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize elimination but make sign errors when adding equations.

For example, they might write \(\mathrm{3x + (-3x + y) = 12 + (-6)}\) as \(\mathrm{3x - 3x + y = 12 - 6}\), getting \(\mathrm{y = 6}\) correctly, but if they mishandle the signs in \(\mathrm{12 + (-6)}\), they might get \(\mathrm{y = 18}\). This may lead them to select Choice C (18).

The Bottom Line:

This problem rewards students who can quickly spot structural patterns (opposite coefficients) and execute clean algebraic manipulation. The key insight is recognizing when elimination is the cleanest path forward.