3x + 21 = 3x + kIn the given equation, k is a constant. The equation has infinitely many solutions....

1. TRANSLATE the problem information

• Given information:
  • Equation: \(3\mathrm{x} + 21 = 3\mathrm{x} + \mathrm{k}\)
  • The equation has infinitely many solutions
  • Need to find the value of k

• What "infinitely many solutions" means: The equation must be true for every possible value of x

2. INFER the solution strategy

• For the equation to be true for all values of x, both sides must be identical after simplification
• Since both sides already have 3x, we need the constant terms to be equal
• Strategy: Subtract 3x from both sides to isolate the constant terms

3. SIMPLIFY by eliminating the variable terms

• Subtract 3x from both sides:
  \(3\mathrm{x} + 21 - 3\mathrm{x} = 3\mathrm{x} + \mathrm{k} - 3\mathrm{x}\)
• This gives us: \(21 = \mathrm{k}\)

4. Verify the answer

• If \(\mathrm{k} = 21\), the equation becomes: \(3\mathrm{x} + 21 = 3\mathrm{x} + 21\)
• This is indeed true for any value of x

Answer: 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "infinitely many solutions" means mathematically

Students might think this means "solve for x" and attempt to isolate x, getting:
\(3\mathrm{x} + 21 = 3\mathrm{x} + \mathrm{k}\)
\(21 = \mathrm{k}\) (after subtracting 3x)
\(0 = 0\) (if they try to continue solving for x)

This confusion about the goal leads to uncertainty and guessing rather than recognizing that \(\mathrm{k} = 21\) is the answer.

Second Most Common Error:

Conceptual confusion: Mixing up "infinitely many solutions" with "no solutions"

Students might think that since the x-terms are identical on both sides, there's no solution, leading them to believe k should make the equation impossible to solve. This misconception causes them to look for a value of k that creates a contradiction.

The Bottom Line:

The key insight is recognizing that "infinitely many solutions" means the equation must be an identity - true for all x-values - rather than trying to solve for a specific x-value.