A piece of wire with a length of 32 inches is cut into two parts. One part has a length...

1. TRANSLATE the problem information

• Given information:
  • Wire total length: 32 inches
  • Two parts: \(\mathrm{x}\) inches and \(\mathrm{y}\) inches
  • Relationship: \(\mathrm{x}\) is 4 more than 3 times \(\mathrm{y}\)
• What this tells us:
  • \(\mathrm{x + y = 32}\) (parts add up to whole)
  • \(\mathrm{x = 3y + 4}\) (relationship between x and y)

2. INFER the solution approach

• We have two equations and two unknowns - this is a system of equations
• Since one equation already isolates x, substitution is the most efficient method
• Substitute the expression for x into the first equation

3. SIMPLIFY by substitution

• Substitute \(\mathrm{x = 3y + 4}\) into \(\mathrm{x + y = 32}\):
  \(\mathrm{(3y + 4) + y = 32}\)
• Combine like terms:
  \(\mathrm{4y + 4 = 32}\)
• Solve for y:
  \(\mathrm{4y = 28}\)
  \(\mathrm{y = 7}\)

4. SIMPLIFY to find x

• Substitute \(\mathrm{y = 7}\) back into \(\mathrm{x = 3y + 4}\):
  \(\mathrm{x = 3(7) + 4 = 21 + 4 = 25}\)

Answer: 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly interpret "x is 4 more than 3 times y" as \(\mathrm{x = 4 + 3y}\) instead of recognizing the proper order of operations should give \(\mathrm{x = 3y + 4}\). While algebraically equivalent, this can create confusion in subsequent steps.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when combining like terms or isolating variables. For example, they might write \(\mathrm{3y + 4 + y = 32}\) but then incorrectly combine to get \(\mathrm{3y + 5 = 32}\), leading to \(\mathrm{y = 9}\) and \(\mathrm{x = 31}\).

This may lead them to select an incorrect answer or abandon the systematic approach.

The Bottom Line:

This problem tests whether students can accurately convert word relationships into mathematical equations and then systematically solve the resulting system. The key challenge is maintaining precision in both translation and algebraic manipulation.