4x + 6 = 18 Which equation has the same solution as the given equation?...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{4x + 6 = 18}\)
• Need to find: Which equation has the same solution

2. INFER the approach

• Since we want the same solution, we need an equivalent equation
• Equivalent equations are created by performing the same valid operation to both sides
• The goal is to isolate the 4x term on one side

3. SIMPLIFY by applying properties of equality

• Start with: \(\mathrm{4x + 6 = 18}\)
• Subtract 6 from both sides: \(\mathrm{4x + 6 - 6 = 18 - 6}\)
• Simplify the arithmetic: \(\mathrm{4x = 12}\)

4. INFER the final answer

• The equation \(\mathrm{4x = 12}\) is equivalent to our original equation
• Looking at the choices, this matches Choice C

Answer: C. 4x = 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students perform the wrong operation instead of subtraction

Instead of subtracting 6 from both sides, they might:

• Add 6 to both sides:
  \(\mathrm{4x + 6 + 6 = 18 + 6}\)
  \(\mathrm{4x = 30}\) (not among choices, leads to confusion)
• Add 6 to the right side only:
  \(\mathrm{4x + 6 = 18 + 6}\)
  \(\mathrm{4x + 6 = 24}\)
  \(\mathrm{4x = 18}\) (still not matching choices)

More commonly, they add 6 and 18: \(\mathrm{4x = 24}\), leading them to select Choice B (\(\mathrm{4x = 24}\))

Second Most Common Error:

Conceptual confusion about operations: Students multiply instead of subtract

They see \(\mathrm{4x + 6 = 18}\) and think they should multiply \(\mathrm{18 \times 6 = 108}\), leading them to select Choice A (\(\mathrm{4x = 108}\))

The Bottom Line:

This problem tests whether students can systematically apply inverse operations to create equivalent equations. The key insight is recognizing that "undoing" the +6 requires subtracting 6 from both sides.