If (x, y) is the solution to the given system of equations, what is the value of y?x + y...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x + y = 20}\)
  • \(\mathrm{2(x + y) + 3y = 85}\)
• Find: the value of y

2. INFER the most efficient approach

• Notice that the first equation gives us \(\mathrm{x + y = 20}\)
• The second equation contains the expression \(\mathrm{(x + y)}\), which we can substitute directly
• This substitution approach avoids having to solve for x first

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{x + y = 20}\) into the second equation:
  \(\mathrm{2(20) + 3y = 85}\)
• Simplify: \(\mathrm{40 + 3y = 85}\)
• Subtract 40 from both sides: \(\mathrm{3y = 45}\)
• Divide by 3: \(\mathrm{y = 15}\)

4. Verify the solution

• If \(\mathrm{y = 15}\), then \(\mathrm{x = 20 - 15 = 5}\)
• Check in second equation: \(\mathrm{2(5 + 15) + 3(15) = 40 + 45 = 85}\) ✓

Answer: B. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve for x first by using elimination or more complex substitution methods, making the problem unnecessarily difficult. They miss the key insight that x + y appears as a complete unit in the second equation and can be directly substituted.

This leads to more complex algebraic work, increasing chances of calculation errors, and may cause confusion that leads to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the substitution strategy but make arithmetic errors when solving \(\mathrm{3y = 45}\) or during verification. Common mistakes include dividing incorrectly (getting \(\mathrm{y = 5}\) instead of \(\mathrm{y = 15}\)) or making sign errors.

This may lead them to select Choice A (10) if they confuse their x and y values during verification.

The Bottom Line:

The elegance of this problem lies in recognizing that you don't need to find individual values of x and y first - the structure of the equations allows for direct substitution of the entire expression \(\mathrm{x + y = 20}\).