Question: 9x - 7 = 5x + 21 What is the value of x?...

1. INFER the solving strategy

• Given: \(\mathrm{9x - 7 = 5x + 21}\)
• Strategy: Collect all x-terms on one side and all constants on the other side
• This will allow us to isolate x

2. SIMPLIFY by moving x-terms to the left side

• Subtract 5x from both sides: \(\mathrm{9x - 5x - 7 = 5x - 5x + 21}\)
• This gives us: \(\mathrm{4x - 7 = 21}\)

3. SIMPLIFY by moving constants to the right side

• Add 7 to both sides: \(\mathrm{4x - 7 + 7 = 21 + 7}\)
• This gives us: \(\mathrm{4x = 28}\)

4. SIMPLIFY by isolating x

• Divide both sides by 4: \(\mathrm{x = 28 \div 4 = 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when moving terms across the equals sign

Students might write \(\mathrm{9x - 7 = 5x + 21}\), then incorrectly get \(\mathrm{9x + 7 = 5x - 21}\) (flipping signs incorrectly), leading to \(\mathrm{4x = -28}\) and \(\mathrm{x = -7}\). This fundamental misunderstanding of the properties of equality leads to the wrong answer.

Second Most Common Error:

Poor SIMPLIFY reasoning: Arithmetic mistakes in combining like terms

Students correctly identify the strategy but make calculation errors like \(\mathrm{9x - 5x = 5x}\) (instead of 4x) or \(\mathrm{21 + 7 = 27}\) (instead of 28). These computational slips lead to incorrect final answers even with correct approach.

The Bottom Line:

Linear equations require both strategic thinking (knowing to collect like terms) and careful execution of algebraic steps. Most errors stem from either misunderstanding how to properly move terms across the equals sign or making simple arithmetic mistakes during the solving process.