y = 1/3x - 14 y = -x + 18 The solution to the given system of equations is \(\mathrm{(x,...

1. INFER the solution strategy

• Given information:
  • Two equations: \(\mathrm{y = \frac{1}{3}x - 14}\) and \(\mathrm{y = -x + 18}\)
  • Both equations are solved for y
• Key insight: Since both expressions equal y, we can set them equal to each other to eliminate y and solve for x

2. SIMPLIFY by setting up the equation

• Set the right sides equal: \(\mathrm{\frac{1}{3}x - 14 = -x + 18}\)
• This eliminates y and gives us one equation with one unknown

3. SIMPLIFY to isolate x

• Add 14 to both sides: \(\mathrm{\frac{1}{3}x = -x + 32}\)
• Add x to both sides: \(\mathrm{\frac{1}{3}x + x = 32}\)
• Combine like terms: \(\mathrm{\frac{1}{3}x + \frac{3}{3}x = \frac{4}{3}x = 32}\)
• Multiply both sides by 3/4: \(\mathrm{x = 32 \times \frac{3}{4} = 24}\)

Answer: 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes when working with fractions, particularly when adding \(\mathrm{\frac{1}{3}x + x}\).

Many students incorrectly compute \(\mathrm{\frac{1}{3}x + x}\) as \(\mathrm{\frac{2}{3}x}\) instead of recognizing that \(\mathrm{x = \frac{3}{3}x}\), so the sum is \(\mathrm{\frac{4}{3}x}\). This leads to the wrong equation \(\mathrm{\frac{2}{3}x = 32}\), giving \(\mathrm{x = 48}\). While this specific wrong answer may not match any given choices, the fraction errors cause confusion and lead to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors or forget to apply operations to both sides consistently.

For example, when moving terms across the equals sign, they might incorrectly get \(\mathrm{\frac{1}{3}x = -x - 32}\) instead of \(\mathrm{\frac{1}{3}x = -x + 32}\), or make similar mistakes that compound through the solution. This leads to completely incorrect values and abandoning systematic solution for guessing.

The Bottom Line:

This problem requires careful fraction arithmetic and systematic algebraic manipulation. Success depends on methodically applying the same operation to both sides while correctly handling fraction addition.