\(7(2\mathrm{x} - 3) = 63\) Which equation has the same solution as the given equation?...

1. TRANSLATE the problem requirements

• Given: \(7(2\mathrm{x} - 3) = 63\)
• Find: Which equation has the same solution (is equivalent)

2. INFER the most efficient approach

• Since we have 7 multiplying the entire expression \((2\mathrm{x} - 3)\), dividing both sides by 7 will isolate \((2\mathrm{x} - 3)\)
• This creates a simpler equivalent equation

3. SIMPLIFY by dividing both sides by 7

• \(7(2\mathrm{x} - 3) ÷ 7 = 63 ÷ 7\)
• \((2\mathrm{x} - 3) = 9\)
• This gives us: \(2\mathrm{x} - 3 = 9\)

4. INFER the answer from the choices

• Choice A shows: \(2\mathrm{x} - 3 = 9\)
• This exactly matches our simplified equivalent equation

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think they must distribute the 7 first, leading to \(14\mathrm{x} - 21 = 63\)

Instead of recognizing that dividing by 7 creates the simplest equivalent form, they distribute to get \(14\mathrm{x} - 21 = 63\), then mistakenly think this should match one of the answer choices directly. They might see Choice C \((2\mathrm{x} - 21 = 63)\) and think it looks similar, not noticing that the coefficient of x is wrong.

This may lead them to select Choice C \((2\mathrm{x} - 21 = 63)\)

Second Most Common Error:

Conceptual confusion about equivalent equations: Students don't understand what "same solution" means

They might solve the original equation correctly to get \(\mathrm{x} = 6\), then substitute this value into each answer choice to see which one works, rather than recognizing they need to find algebraically equivalent forms. This approach is more time-consuming and prone to arithmetic errors.

This leads to confusion and potentially random guessing between choices.

The Bottom Line:

The key insight is that equivalent equations can be created through valid algebraic operations like dividing both sides by the same non-zero number. The most efficient approach is often the one that creates the simplest form.