The equation 14x - 5 = 23 has a real solution x. What value of k makes the equation 3x...

1. INFER what "same solution" means

• Key insight: If two equations have the same solution, then the same x-value must work in both equations
• Strategy: Solve the first equation to find x, then use that x-value in the second equation

2. SIMPLIFY the first equation to find x

• Start with: \(14\mathrm{x} - 5 = 23\)
• Add 5 to both sides: \(14\mathrm{x} = 28\)
• Divide by 14: \(\mathrm{x} = 2\)

3. SIMPLIFY the second equation using the known solution

• We know \(\mathrm{x} = 2\) must satisfy \(3\mathrm{x} + \mathrm{k} = 7\)
• Substitute \(\mathrm{x} = 2\): \(3(2) + \mathrm{k} = 7\)
• Simplify: \(6 + \mathrm{k} = 7\)
• Solve for k: \(\mathrm{k} = 1\)

Answer: C (1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't understand what "same solution" means and try to solve the equations simultaneously as a system, or attempt to set the equations equal to each other.

Instead of finding \(\mathrm{x} = 2\) from the first equation, they might try something like:
\(14\mathrm{x} - 5 = 3\mathrm{x} + \mathrm{k}\), leading to confusion about having two unknowns.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students solve the first equation incorrectly due to arithmetic errors.

For example, they might get \(14\mathrm{x} = 18\) (instead of 28) when adding 5 to both sides, leading to \(\mathrm{x} = 18/14\). Then substituting this wrong x-value gives an incorrect k-value.

This may lead them to select Choice A (-2) or other incorrect answers.

The Bottom Line:

The key insight is recognizing that "same solution" is a bridge concept - you use the solution from one equation as input to find the missing parameter in the other equation. Students who miss this connection often overcomplicate the problem.