The system of equations below has a solution \(\mathrm{(x, y)}\). 4x + 3y = 5 2x + y = 3...

1. INFER the solution strategy

• Given system:
  • \(\mathrm{4x + 3y = 5}\)
  • \(\mathrm{2x + y = 3}\)
• We need to find \(\mathrm{x - y}\), so we need both x and y values first
• The second equation is simpler - it's easier to solve for y in terms of x from there

2. SIMPLIFY to express y in terms of x

From \(\mathrm{2x + y = 3}\):

\(\mathrm{y = 3 - 2x}\)

3. SIMPLIFY using substitution

Substitute \(\mathrm{y = 3 - 2x}\) into the first equation:

\(\mathrm{4x + 3(3 - 2x) = 5}\)

Distribute carefully:

\(\mathrm{4x + 9 - 6x = 5}\)

Combine like terms:

\(\mathrm{-2x + 9 = 5}\)

Subtract 9 from both sides:

\(\mathrm{-2x = -4}\)

Divide by -2:

\(\mathrm{x = 2}\)

4. SIMPLIFY to find y

\(\mathrm{y = 3 - 2x = 3 - 2(2) = 3 - 4 = -1}\)

5. Calculate the final answer

\(\mathrm{x - y = 2 - (-1) = 2 + 1 = 3}\)

Answer: D (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Distribution and sign errors when expanding \(\mathrm{4x + 3(3 - 2x) = 5}\)

Students often make mistakes like:

• Incorrect distribution: \(\mathrm{4x + 9 + 6x = 5}\) (forgetting the negative sign)
• Arithmetic errors: \(\mathrm{-2x = -4 \rightarrow x = -2}\) (division error)
• Sign confusion when subtracting negative numbers: \(\mathrm{2 - (-1) = 1}\)

These algebraic errors lead to wrong x and y values, causing them to select Choice A (-2), Choice B (1), or Choice C (2).

Second Most Common Error:

Conceptual confusion: Calculating \(\mathrm{x + y}\) instead of \(\mathrm{x - y}\)

Some students rush through the final step and compute \(\mathrm{x + y = 2 + (-1) = 1}\), leading them to select Choice B (1).

The Bottom Line:

This problem tests careful algebraic execution more than conceptual understanding. Success requires systematic attention to signs, distribution, and arithmetic—areas where rushing often leads to preventable errors.