x + 3y = 29 3y = 11 The solution to the given system of equations is \(\mathrm{(x, y)}\). What...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x + 3y = 29}\)
  • \(\mathrm{3y = 11}\)
• Find: The value of x

2. INFER the most efficient approach

• Notice that the second equation already gives us the value of 3y
• We can substitute \(\mathrm{3y = 11}\) directly into the first equation
• This avoids the extra step of solving for y first

3. SIMPLIFY by substitution

• Substitute \(\mathrm{3y = 11}\) into \(\mathrm{x + 3y = 29}\):
  \(\mathrm{x + 11 = 29}\)
• Subtract 11 from both sides:
  \(\mathrm{x = 29 - 11 = 18}\)

Answer: 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct substitution opportunity and instead try to solve for y first from \(\mathrm{3y = 11}\), getting \(\mathrm{y = 11/3}\). They then substitute this fraction back into the first equation, leading to more complex arithmetic: \(\mathrm{x + 3(11/3) = 29}\). While this approach works, the unnecessary complexity increases chances for calculation errors and wastes time.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x + 11 = 29}\) but make arithmetic errors when solving, such as \(\mathrm{x = 29 + 11 = 40}\) instead of \(\mathrm{x = 29 - 11 = 18}\). This leads to an incorrect final answer.

The Bottom Line:

This problem rewards students who can spot the elegant shortcut. The key insight is recognizing that when one equation gives you an expression (\(\mathrm{3y = 11}\)), you can use that entire expression as a building block rather than breaking it down further.