If \(\mathrm{(x, y)}\) is the solution to the given system of equations, what is the value of y?3x + y...

1. TRANSLATE the problem information

• Given information:
  • System: \(\mathrm{3x + y = 29}\) and \(\mathrm{x = 2}\)
  • Need to find: the value of y

2. INFER the solving strategy

• Since \(\mathrm{x = 2}\) is already provided, we don't need to solve a complex system
• We can substitute \(\mathrm{x = 2}\) directly into the first equation to find y
• This is much simpler than trying elimination or other system-solving methods

3. SIMPLIFY through substitution and solving

• Substitute \(\mathrm{x = 2}\) into \(\mathrm{3x + y = 29}\):
  • \(\mathrm{3(2) + y = 29}\)
  • \(\mathrm{6 + y = 29}\)
• Isolate y by subtracting 6 from both sides:
  • \(\mathrm{y = 29 - 6}\)
  • \(\mathrm{y = 23}\)

Answer: 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{x = 2}\) is already the solution for x, so they try to "solve the system" using elimination or other complex methods when simple substitution is all that's needed.

This leads to unnecessary work and potential confusion, causing them to second-guess their straightforward answer or make computational errors in overly complex approaches.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the substitution process, such as calculating \(\mathrm{3(2) = 5}\) instead of \(\mathrm{6}\), or computing \(\mathrm{29 - 6 = 22}\) instead of \(\mathrm{23}\).

This leads them to get an incorrect final answer even though their approach was correct.

The Bottom Line:

This problem tests whether students can recognize when they've been given a direct shortcut (\(\mathrm{x = 2}\)) rather than needing to solve a full system, and whether they can execute basic substitution without arithmetic errors.