\(3(\mathrm{kx} + 13) = \frac{48}{17}\mathrm{x} + 36\)In the given equation, k is a constant. The equation has no solution. What...

1. TRANSLATE the problem information

• Given: \(3(\mathrm{kx} + 13) = \frac{48}{17}\mathrm{x} + 36\)
• Key condition: The equation has no solution
• Find: The value of k

2. INFER what "no solution" means

• A linear equation \(\mathrm{ax} + \mathrm{b} = \mathrm{cx} + \mathrm{d}\) has no solution when:
  • The coefficients of x are equal (\(\mathrm{a} = \mathrm{c}\))
  • BUT the constant terms are different (\(\mathrm{b} \neq \mathrm{d}\))
• This creates a contradiction like "5 = 7" which is impossible

3. SIMPLIFY the equation to standard form

• Expand the left side: \(3(\mathrm{kx} + 13) = 3\mathrm{kx} + 39\)
• Rewrite: \(3\mathrm{kx} + 39 = \frac{48}{17}\mathrm{x} + 36\)
• Now we can compare coefficients and constants

4. INFER the condition for no solution

• For no solution, coefficients must be equal:
  \(3\mathrm{k} = \frac{48}{17}\)
• Solve for k: \(\mathrm{k} = \frac{48}{17} \div 3 = \frac{48}{51} = \frac{16}{17}\)

5. SIMPLIFY the fraction

• \(\frac{48}{51} = \frac{16}{17}\) (dividing both numerator and denominator by 3)

Answer: \(\frac{16}{17}\) (or \(0.941, .9411, .9412\) in decimal form)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Misunderstanding what "no solution" means algebraically

Students often think "no solution" means the equation equals zero or that they should solve for x. They might set up \(3(\mathrm{kx} + 13) = \frac{48}{17}\mathrm{x} + 36 = 0\), leading to incorrect algebraic manipulation. This confusion about the fundamental concept of "no solution" causes them to abandon systematic solution and guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when handling fractions

Students correctly identify that coefficients must be equal (\(3\mathrm{k} = \frac{48}{17}\)) but make errors in the division: \(\mathrm{k} = \frac{48}{17} \div 3\). Common mistakes include forgetting to multiply by the reciprocal or incorrectly simplifying \(\frac{48}{51}\). This leads them to arrive at incorrect values that don't match any of the given answer forms.

The Bottom Line:

This problem tests whether students truly understand the algebraic meaning of "no solution" rather than just mechanical equation-solving. The key insight is recognizing that "no solution" creates specific structural requirements for the equation's coefficients and constants.