\(10-(\mathrm{n}+4)=3\) What value of n is the solution to the given equation? -{3} 3 11 9...

1. INFER the solution strategy

• We have a linear equation with parentheses: \(\mathrm{10-(n+4)=3}\)
• Strategy: First distribute the negative sign, then combine like terms, then isolate the variable

2. SIMPLIFY by distributing the negative sign

• \(\mathrm{10-(n+4)}\) becomes \(\mathrm{10-n-4}\)
• Remember: The negative sign applies to everything inside the parentheses
• Our equation is now: \(\mathrm{10-n-4=3}\)

3. SIMPLIFY by combining like terms

• On the left side: \(\mathrm{10-4=6}\)
• Our equation becomes: \(\mathrm{6-n=3}\)

4. SIMPLIFY to isolate the variable term

• Subtract 6 from both sides: \(\mathrm{6-n-6=3-6}\)
• This gives us: \(\mathrm{-n=-3}\)

5. SIMPLIFY to solve for n

• Since we have \(\mathrm{-n=-3}\), multiply both sides by -1
• Final result: \(\mathrm{n=3}\)

Answer: B) 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign error when distributing the negative sign

Students often write \(\mathrm{10-(n+4)}\) as \(\mathrm{10-n+4}\) instead of \(\mathrm{10-n-4}\). They forget that the negative sign must be distributed to both terms inside the parentheses.

Following this error path:

\(\mathrm{10-n+4=3}\)

\(\mathrm{14-n=3}\)

\(\mathrm{-n=-11}\)

\(\mathrm{n=11}\)

This may lead them to select Choice C (11)

Second Most Common Error:

Weak SIMPLIFY execution: Forgetting the final step to solve for n

Students correctly work through the problem until they reach \(\mathrm{-n=-3}\), but then stop or incorrectly think this means \(\mathrm{n=-3}\). They forget to multiply both sides by -1 to get the positive value of n.

This may lead them to select Choice A (-3)

The Bottom Line:

This problem tests careful execution of algebraic procedures, particularly managing negative signs. Success requires systematic application of distribution and solving techniques without rushing through the sign changes.