Question:A linear relationship is given by the equation y = 3x - 7. If the value of y is 14,...

1. TRANSLATE the problem information

• Given information:
  • Linear equation: \(\mathrm{y = 3x - 7}\)
  • Specific value: \(\mathrm{y = 14}\)
  • Find: the value of x

2. TRANSLATE the substitution approach

• Since we know \(\mathrm{y = 14}\), we can substitute this value directly into our linear equation
• This transforms \(\mathrm{y = 3x - 7}\) into \(\mathrm{14 = 3x - 7}\)

3. SIMPLIFY to isolate the variable term

• Current equation: \(\mathrm{14 = 3x - 7}\)
• Add 7 to both sides: \(\mathrm{14 + 7 = 3x - 7 + 7}\)
• This gives us: \(\mathrm{21 = 3x}\)

4. SIMPLIFY to solve for x

• Current equation: \(\mathrm{21 = 3x}\)
• Divide both sides by 3: \(\mathrm{21 ÷ 3 = 3x ÷ 3}\)
• Final result: \(\mathrm{7 = x}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when moving terms across the equals sign, particularly forgetting that subtracting 7 becomes adding 7 when moved to the other side.

For example, they might write: \(\mathrm{14 - 7 = 3x}\) instead of \(\mathrm{14 + 7 = 3x}\), leading to \(\mathrm{7 = 3x}\) and then \(\mathrm{x = 7/3 ≈ 2.33}\). This leads to confusion since the answer should be an integer.

Second Most Common Error:

Arithmetic errors in SIMPLIFY: Students correctly set up the equation but make calculation mistakes, such as \(\mathrm{14 + 7 = 20}\) instead of \(\mathrm{21}\), or \(\mathrm{21 ÷ 3 = 6}\) instead of \(\mathrm{7}\).

This causes them to get a wrong integer answer and second-guess their work.

The Bottom Line:

Linear equation problems are straightforward conceptually, but students often stumble on the mechanical execution of algebraic operations, especially sign changes and basic arithmetic under the pressure of solving systematically.