Question:If 5n - 2 = 18, what is the value of 10n - 7?This is a fill-in-the-blank question.

1. INFER the problem structure

• Given information:
  • Equation: \(5\mathrm{n} - 2 = 18\)
  • Need to find: \(10\mathrm{n} - 7\)

• This problem gives us one equation with variable n, then asks for the value of a different expression containing n

2. INFER the solving approach

• Strategy: Solve the given equation for n, then substitute this value into the expression we need to evaluate
• This is the most straightforward approach for this type of problem

3. SIMPLIFY the given equation to find n

• Start with: \(5\mathrm{n} - 2 = 18\)
• Add 2 to both sides: \(5\mathrm{n} = 20\)
• Divide both sides by 5: \(\mathrm{n} = 4\)

4. SIMPLIFY the target expression using our found value

• Expression to evaluate: \(10\mathrm{n} - 7\)
• Substitute \(\mathrm{n} = 4\): \(10(4) - 7\)
• Calculate: \(40 - 7 = 33\)

Answer: 33

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the multi-step calculations.

Common mistakes include:

• Getting \(5\mathrm{n} = 18\) (forgetting to add 2)
• Calculating \(\mathrm{n} = 3.6\) (dividing 18 by 5 instead of 20 by 5)
• Making errors in the final calculation: \(10(4) - 7 = 47\) (adding instead of subtracting) or getting 33 wrong through other arithmetic mistakes

These calculation errors can lead to various incorrect answers depending on where the mistake occurred.

Second Most Common Error:

Poor INFER reasoning: Students attempt to manipulate \(10\mathrm{n} - 7\) directly without first solving for n.

They might try to set \(10\mathrm{n} - 7\) equal to something or attempt algebraic manipulation without establishing the value of n first. This leads to confusion about how to proceed systematically, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students can follow a systematic two-step approach: solve for the variable, then substitute. The multiple arithmetic steps create opportunities for calculation errors, while the indirect nature (finding one expression given information about another) requires clear strategic thinking.