The equation 46 = 2x + 2y gives the perimeter of a rectangular rug that has length x, in feet,...

1. TRANSLATE the problem information

• Given information:
  • Perimeter equation: \(46 = 2\mathrm{x} + 2\mathrm{y}\)
  • \(\mathrm{x}\) represents length in feet
  • \(\mathrm{y}\) represents width in feet
  • Width is 8 feet (so \(\mathrm{y} = 8\))
• What we need to find: the length (\(\mathrm{x}\) value)

2. INFER the solution approach

• Since we know the width (\(\mathrm{y} = 8\)) and have an equation with both \(\mathrm{x}\) and \(\mathrm{y}\), we can substitute the known value to solve for the unknown length
• Strategy: Replace \(\mathrm{y}\) with 8 in the equation, then solve for \(\mathrm{x}\)

3. SIMPLIFY through substitution and algebraic steps

• Substitute \(\mathrm{y} = 8\) into \(46 = 2\mathrm{x} + 2\mathrm{y}\):
  \(46 = 2\mathrm{x} + 2(8)\)
  \(46 = 2\mathrm{x} + 16\)

• Subtract 16 from both sides:
  \(46 - 16 = 2\mathrm{x}\)
  \(30 = 2\mathrm{x}\)

• Divide both sides by 2:
  \(\mathrm{x} = 15\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they should substitute \(\mathrm{y} = 8\) into the given equation. Instead, they might try to use a different approach like setting up a system of equations or become confused about what to do with the given perimeter equation. This leads to confusion and abandoning systematic solution, often resulting in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the substitution but make arithmetic errors during the solving process. For example, they might incorrectly compute \(46 - 16 = 20\) instead of 30, leading them to get \(\mathrm{x} = 10\) instead of \(\mathrm{x} = 15\). This leads to selecting an incorrect numerical answer.

The Bottom Line:

This problem tests whether students can connect a real-world constraint (given width) with an algebraic equation through substitution. The key insight is recognizing that you have everything needed to solve for the unknown variable once you substitute the known value.