-{3x + 21px = 84}In the given equation, p is a constant. The equation has no solution. What is the...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(-3\mathrm{x} + 21\mathrm{px} = 84\)
  • p is a constant
  • The equation has no solution for x
• Need to find: The value of p

2. INFER when a linear equation has no solution

• A linear equation has no solution when we get a false statement like \(0 = 84\)
• This happens when the coefficient of the variable equals zero but the constant term doesn't
• Strategy: Factor out x and find when its coefficient equals zero

3. SIMPLIFY by factoring the left side

• Factor out x: \(-3\mathrm{x} + 21\mathrm{px} = \mathrm{x}(-3 + 21\mathrm{p})\)
• The equation becomes: \(\mathrm{x}(-3 + 21\mathrm{p}) = 84\)
• The coefficient of x is \((-3 + 21\mathrm{p})\)

4. INFER the condition for no solution

• For no solution: coefficient of x = 0, but right side ≠ 0
• Since \(84 ≠ 0\), we need: \(-3 + 21\mathrm{p} = 0\)
• This would make the equation: \(\mathrm{x}(0) = 84\), or \(0 = 84\) (false!)

5. SIMPLIFY to solve for p

• From \(-3 + 21\mathrm{p} = 0\):
• Add 3 to both sides: \(21\mathrm{p} = 3\)
• Divide by 21: \(\mathrm{p} = \frac{3}{21} = \frac{1}{7}\)

Answer: B. 1/7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't understand what "no solution" means for a linear equation.

Many students try to solve the equation directly by collecting like terms: \((-3 + 21\mathrm{p})\mathrm{x} = 84\), then attempt \(\mathrm{x} = \frac{84}{(-3 + 21\mathrm{p})}\). They might substitute answer choices to see which gives a "weird" result, but miss that the denominator being zero creates the no-solution condition. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students factor correctly but make algebraic mistakes when solving \(-3 + 21\mathrm{p} = 0\).

Common errors include: \(21\mathrm{p} = -3\) (forgetting to add 3), or \(\mathrm{p} = \frac{3}{21} = \frac{1}{3}\) (calculation error). This may lead them to select Choice C (1/4) as the closest to their incorrect calculation.

The Bottom Line:

This problem tests conceptual understanding of when linear equations fail to have solutions, not just algebraic manipulation. Students must recognize that "no solution" occurs when the variable's coefficient becomes zero while the constant remains non-zero.