If \(2(\mathrm{x} - 5) + 3(\mathrm{x} - 5) = 10\), what is the value of x - 5?

1. INFER the structure of the equation

Looking at \(2(\mathrm{x} - 5) + 3(\mathrm{x} - 5) = 10\), I notice both terms on the left side contain the same expression: \((\mathrm{x} - 5)\).

This means I have like terms that can be combined, or I can factor out the common factor \((\mathrm{x} - 5)\).

2. SIMPLIFY by combining like terms

• Factor out \((\mathrm{x} - 5)\): \(2(\mathrm{x} - 5) + 3(\mathrm{x} - 5) = (2 + 3)(\mathrm{x} - 5) = 5(\mathrm{x} - 5)\)
• The equation becomes: \(5(\mathrm{x} - 5) = 10\)

3. SIMPLIFY to solve for x - 5

• Divide both sides by 5: \(\mathrm{x} - 5 = 10 \div 5 = 2\)
• Notice the question asks for \(\mathrm{x} - 5\), which is exactly what we found!

Answer: A. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that both terms contain the same factor \((\mathrm{x} - 5)\) that can be combined or factored out.

Instead, students might try to distribute first: \(2\mathrm{x} - 10 + 3\mathrm{x} - 15 = 10\), then \(5\mathrm{x} - 25 = 10\), so \(5\mathrm{x} = 35\), and \(\mathrm{x} = 7\). But the question asks for \(\mathrm{x} - 5\), not \(\mathrm{x}\), so they'd need to calculate \(7 - 5 = 2\). However, if they miss this final step, they select the value of \(\mathrm{x}\).

This may lead them to select Choice C (7).

Second Most Common Error:

Poor SIMPLIFY execution: After correctly getting \(5(\mathrm{x} - 5) = 10\), making an arithmetic error in the final step.

For example, subtracting 5 from both sides instead of dividing: \(5(\mathrm{x} - 5) - 5 = 10 - 5\), leading to some incorrect manipulation that might yield \(\mathrm{x} - 5 = 5\).

This may lead them to select Choice B (5).

The Bottom Line:

The key insight is recognizing the structure - when you see the same expression repeated with different coefficients, you can combine them just like you would combine \(2\mathrm{x} + 3\mathrm{x} = 5\mathrm{x}\).