6x + k = 6x + 5 In the given equation, k is a constant. If the equation has infinitely...

1. INFER what 'infinitely many solutions' means

• Given information:
  • Equation: \(\mathrm{6x + k = 6x + 5}\)
  • The equation has infinitely many solutions
  • k is a constant we need to find

• Key insight: For a linear equation to have infinitely many solutions, both sides must be identical. This means the equation must reduce to something like '\(\mathrm{a = a}\)' which is always true.

2. SIMPLIFY by eliminating the variable terms

• Since both sides have 6x, subtract 6x from both sides:
  \(\mathrm{6x + k - 6x = 6x + 5 - 6x}\)
  \(\mathrm{k = 5}\)

3. Verify the solution

• If \(\mathrm{k = 5}\), our equation becomes: \(\mathrm{6x + 5 = 6x + 5}\)
• This is true for ANY value of x, confirming infinitely many solutions

Answer: \(\mathrm{5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what 'infinitely many solutions' means for a linear equation

Students might think that 'infinitely many solutions' means k could be any number, or they might confuse this with 'no solution' scenarios. Without grasping that infinitely many solutions requires an identity (both sides exactly the same), they can't determine the specific value k must have.

This leads to confusion and guessing rather than systematic problem-solving.

The Bottom Line:

The key challenge is recognizing that 'infinitely many solutions' has a very specific meaning - the equation must be an identity where both sides are identical. Once students understand this concept, the algebraic manipulation is straightforward.