48x - 64y = 48y + 24ry = 1/3 - 12xIn the given system of equations, r is a constant....

1. TRANSLATE the problem requirements

• Given information:
  • System: \(\mathrm{48x - 64y = 48y + 24}\) and \(\mathrm{ry = \frac{1}{3} - 12x}\)
  • System has no solution
• What this tells us: We need to find the value of r that makes this system have no solution

2. INFER the mathematical condition

• A system has no solution when the lines are parallel but distinct
• Lines are parallel when their coefficients are proportional
• We need both equations in standard form \(\mathrm{Ax + By = C}\) to compare coefficients

3. TRANSLATE both equations to standard form

• First equation: \(\mathrm{48x - 64y = 48y + 24}\)
  Subtract 48y from both sides: \(\mathrm{48x - 112y = 24}\)
• Second equation: \(\mathrm{ry = \frac{1}{3} - 12x}\)
  Add 12x to both sides: \(\mathrm{12x + ry = \frac{1}{3}}\)

4. INFER the proportionality requirement

• Standard forms: \(\mathrm{48x - 112y = 24}\) and \(\mathrm{12x + ry = \frac{1}{3}}\)
• For parallel lines: coefficient ratios must be equal
• x-coefficient ratio: \(\mathrm{\frac{48}{12} = 4}\)
• y-coefficient ratio must equal 4: \(\mathrm{\frac{-112}{r} = 4}\)

5. SIMPLIFY to solve for r

• From \(\mathrm{\frac{-112}{r} = 4}\):
  \(\mathrm{-112 = 4r}\)
  \(\mathrm{r = -28}\)

6. INFER verification of "no solution"

• Check that lines are distinct (not identical):
• Constant ratio: \(\mathrm{\frac{24}{\frac{1}{3}} = 72}\)
• Since \(\mathrm{72 \neq 4}\), the lines are parallel but distinct ✓

Answer: -28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what "no solution" means for a system of equations

Students might try to solve the system directly by substitution or elimination, getting confused when they can't find values for x and y. They don't realize that "no solution" is a condition that tells us something about the relationship between the coefficients.

This leads to confusion and abandoning systematic solution, often resulting in guessing.

Second Most Common Error:

Poor TRANSLATE execution: Incorrectly converting equations to standard form

Students might make algebraic errors when rearranging terms, such as:

• Forgetting to move all terms to one side
• Sign errors when moving terms across the equal sign
• Not combining like terms correctly (48y and -64y)

This may lead them to set up incorrect proportions and calculate a wrong value for r.

The Bottom Line:

The key insight is recognizing that "no solution" is not a dead end—it's actually valuable information that defines a specific relationship between the coefficients. Once you understand that parallel lines create no-solution systems, the algebra becomes straightforward.