The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of x + y?5x =...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{5x = 15}\)
  • \(\mathrm{-4x + y = -2}\)
• What we need: The value of \(\mathrm{x + y}\)

2. INFER the most efficient approach

• Key insight: Since we need \(\mathrm{x + y}\) (not individual values), we can add the equations directly
• Adding the left sides: \(\mathrm{5x + (-4x + y) = x + y}\)
• Adding the right sides: \(\mathrm{15 + (-2) = 13}\)

3. SIMPLIFY the equation addition

• \(\mathrm{5x + (-4x + y) = 15 + (-2)}\)
• \(\mathrm{x + y = 13}\)

Answer: C. 13

Alternative Method:

If you prefer finding individual values first:

1. From \(\mathrm{5x = 15}\): \(\mathrm{x = 3}\)
2. Substitute into \(\mathrm{-4x + y = -2}\): \(\mathrm{-4(3) + y = -2}\), so \(\mathrm{y = 10}\)
3. Therefore \(\mathrm{x + y = 3 + 10 = 13}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they can add equations directly and instead get bogged down trying to solve for each variable individually, leading to unnecessary complexity and potential calculation errors.

This approach still works, but increases chances of arithmetic mistakes that may lead them to select Choice A (-17) or Choice D (17) due to sign errors.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when manipulating the second equation, particularly when dealing with \(\mathrm{-4x + y = -2}\). Common mistake: getting \(\mathrm{y = -14}\) instead of \(\mathrm{y = 10}\), which would give \(\mathrm{x + y = 3 + (-14) = -11}\).

This doesn't directly match any answer choice, leading to confusion and guessing.

The Bottom Line:

This problem rewards strategic thinking - recognizing that you can work directly toward \(\mathrm{x + y}\) rather than finding each variable separately. The direct addition method is both faster and less error-prone.