The average of x and y is 5. Also, 3x + 4y = 28. If \(\mathrm{(x, y)}\) satisfies both statements,...

1. TRANSLATE the average statement into math

• Given information:
  • "The average of x and y is 5" means \(\frac{\mathrm{x + y}}{2} = 5\)
  • Also given: \(\mathrm{3x + 4y = 28}\)
• From the average equation: \(\mathrm{x + y = 10}\)

2. INFER that we have a system of equations

• We now have two equations with two unknowns:
  • \(\mathrm{x + y = 10}\)
  • \(\mathrm{3x + 4y = 28}\)
• This system can be solved using substitution or elimination

3. SIMPLIFY using substitution method

• From \(\mathrm{x + y = 10}\), solve for x: \(\mathrm{x = 10 - y}\)
• Substitute into the second equation:
  \(\mathrm{3(10 - y) + 4y = 28}\)
• Distribute: \(\mathrm{30 - 3y + 4y = 28}\)
• Combine like terms: \(\mathrm{30 + y = 28}\)
• Solve: \(\mathrm{y = -2}\)

4. Verify the solution

• If \(\mathrm{y = -2}\), then \(\mathrm{x = 10 - (-2) = 12}\)
• Check in original equation: \(\mathrm{3(12) + 4(-2) = 36 - 8 = 28}\) ✓

Answer: y = -2 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to convert "the average of x and y is 5" into a mathematical equation. They might write something like "\(\mathrm{x + y = 5}\)" instead of recognizing that average means \(\frac{\mathrm{x + y}}{2} = 5\), which gives \(\mathrm{x + y = 10}\).

This incorrect translation leads to the wrong system:

• \(\mathrm{x + y = 5}\) (incorrect)
• \(\mathrm{3x + 4y = 28}\)

Solving this incorrect system would give \(\mathrm{y = -1}\), leading them to select Choice (D) (-1).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the system but make sign errors during algebraic manipulation. For example, when distributing \(\mathrm{3(10 - y)}\), they might write \(\mathrm{30 + 3y}\) instead of \(\mathrm{30 - 3y}\), or make errors when combining like terms.

These algebraic mistakes can lead to various incorrect values, causing confusion and potentially leading them to select Choice (A) (-4) or Choice (B) (-3) depending on the specific error.

The Bottom Line:

This problem tests whether students can accurately translate everyday language about averages into mathematical equations. The word "average" is the key - students must remember that average requires dividing by the number of values, not just adding them up.