13x = 112 - x What value of x is the solution to the given equation?...

1. INFER the solution strategy

• Looking at \(\mathrm{13x = 112 - x}\), we have x terms on both sides
• Strategy: Collect all x terms on one side to isolate the variable
• Best approach: Add x to both sides to move the -x term

2. SIMPLIFY by adding x to both sides

• \(\mathrm{13x + x = 112 - x + x}\)
• Left side: \(\mathrm{13x + x = 14x}\)
• Right side: \(\mathrm{112 - x + x = 112}\)
• Result: \(\mathrm{14x = 112}\)

3. SIMPLIFY by dividing both sides by 14

• \(\mathrm{14x ÷ 14 = 112 ÷ 14}\)
• \(\mathrm{x = 8}\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students subtract x from both sides instead of adding x, thinking they need to "get rid of" the x on the right side.

Following this incorrect reasoning: \(\mathrm{13x - x = 112 - x - x}\) leads to \(\mathrm{12x = 112 - 2x}\), which creates an even more complex equation with x terms still on both sides. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly add x to both sides but make arithmetic errors when combining like terms or dividing.

Common mistakes include: \(\mathrm{13x + x = 13x}\) (forgetting the coefficient of x is 1) or dividing \(\mathrm{112 ÷ 14}\) incorrectly. This leads to wrong numerical answers.

The Bottom Line:

This problem tests whether students can recognize the most efficient strategy for collecting like terms and then execute basic algebraic operations accurately. The key insight is that adding the variable term to both sides eliminates it from one side completely.