y = 2x - 33y = 5xIn the solution to the system of equations above, what is the value of...

1. INFER the problem structure

• Given information:
  • \(\mathrm{y = 2x - 3}\) (equation 1)
  • \(\mathrm{3y = 5x}\) (equation 2)
• What this tells us: We have a system of two linear equations with two unknowns, and we need to find the value of y.

2. INFER the solution strategy

• Since equation 1 already gives us y in terms of x, substitution is the most direct approach
• We can substitute the expression for y from equation 1 into equation 2

3. SIMPLIFY through substitution

• Substitute \(\mathrm{y = 2x - 3}\) into equation 2:
  \(\mathrm{3(2x - 3) = 5x}\)
• Apply distributive property:
  \(\mathrm{6x - 9 = 5x}\)

4. SIMPLIFY to solve for x

• Combine like terms:
  \(\mathrm{6x - 5x = 9}\)
  \(\mathrm{x = 9}\)

5. SIMPLIFY to find y

• Substitute \(\mathrm{x = 9}\) back into equation 1:
  \(\mathrm{y = 2(9) - 3 = 18 - 3 = 15}\)

Answer: D. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during distribution or when combining like terms.

For example, when distributing \(\mathrm{3(2x - 3)}\), they might get \(\mathrm{6x - 3}\) instead of \(\mathrm{6x - 9}\), or when solving \(\mathrm{6x - 5x = 9}\), they might incorrectly get \(\mathrm{x = -9}\). These calculation errors cascade through the remaining steps, leading them to select Choice B (-9) or other incorrect answers.

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve for x instead of y, or stop after finding \(\mathrm{x = 9}\) without completing the substitution.

This leads them to select Choice C (9), which represents the x-value rather than the requested y-value.

The Bottom Line:

This problem tests whether students can systematically work through a system of equations while maintaining accuracy in algebraic manipulation and staying focused on what the question actually asks for.