x + 2y = 6 x - 2y = 4 The solution to the given system of equations is \(\mathrm{(x,...

1. TRANSLATE the problem information

• Given system:
  • First equation: \(\mathrm{x + 2y = 6}\)
  • Second equation: \(\mathrm{x - 2y = 4}\)
• We need to find the value of x

2. INFER the solution strategy

• Notice that the y terms have opposite coefficients (\(\mathrm{+2y}\) and \(\mathrm{-2y}\))
• Adding these equations will eliminate the y variable completely
• This makes elimination the most efficient approach

3. SIMPLIFY by adding the equations

• Add left sides and right sides together:
  \(\mathrm{(x + 2y) + (x - 2y) = 6 + 4}\)
• Combine like terms:
  \(\mathrm{x + x + 2y - 2y = 10}\)
  \(\mathrm{2x + 0 = 10}\)
  \(\mathrm{2x = 10}\)

4. SIMPLIFY to find x

• Divide both sides by 2:
  \(\mathrm{x = \frac{10}{2} = 5}\)

Answer: B. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the elimination but make arithmetic errors when combining terms or solving the final equation.

They might incorrectly combine terms as "\(\mathrm{x + x + 2y - 2y = x + 0 = x}\)" instead of "\(\mathrm{2x + 0 = 2x}\)", or they might correctly get \(\mathrm{2x = 10}\) but forget to divide by 2 to isolate x.

This may lead them to select Choice D (10) by stopping at the \(\mathrm{2x = 10}\) step, or Choice C (6) through various calculation errors.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that elimination by addition is the most direct approach and instead attempt substitution, which is more complex for this particular system.

They might solve correctly but waste time, or make errors during the more involved substitution process. This leads to confusion and potential guessing among the answer choices.

The Bottom Line:

This problem rewards students who can quickly INFER that the coefficients of y are opposites (\(\mathrm{+2y}\) and \(\mathrm{-2y}\)), making elimination by addition the obvious choice, then carefully SIMPLIFY through the arithmetic without rushing.