What value of x is the solution to the equation 16x + 24 = 24x?

1. INFER the solving strategy

• Given: \(16\mathrm{x} + 24 = 24\mathrm{x}\)
• Goal: Isolate x on one side of the equation
• Strategy: First eliminate x terms from one side, then divide by the remaining coefficient

2. SIMPLIFY by collecting like terms

• Subtract 16x from both sides:
  • Left side: \(16\mathrm{x} + 24 - 16\mathrm{x} = 24\)
  • Right side: \(24\mathrm{x} - 16\mathrm{x} = 8\mathrm{x}\)
• Result: \(24 = 8\mathrm{x}\)

3. SIMPLIFY by isolating x

• Divide both sides by 8:
  • \(24 \div 8 = 3\)
  • \(8\mathrm{x} \div 8 = \mathrm{x}\)
• Result: \(\mathrm{x} = 3\)

Answer: D. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Confusing which coefficient to divide by or making arithmetic errors during the division step.

After correctly getting to \(24 = 8\mathrm{x}\), some students might think "x equals the coefficient divided by the constant" and calculate \(\mathrm{x} = 8/24 = 1/3\) instead of \(\mathrm{x} = 24/8 = 3\). This reverses the division operation.

This may lead them to select Choice C (1/3).

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors or calculation mistakes when combining like terms.

Students might incorrectly subtract terms or make arithmetic errors, leading to equations like \(-8\mathrm{x} = -32\) (giving \(\mathrm{x} = 4\), not among choices) or other incorrect intermediate steps that cause confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically apply inverse operations while maintaining accuracy in basic arithmetic - the foundation of algebraic equation solving.