48x - 72y = 30y + 24ry = 1/3 - 16xIn the given system of equations, r is a constant....

1. TRANSLATE both equations to standard form

• Given equations:
  • \(\mathrm{48x - 72y = 30y + 24}\)
  • \(\mathrm{ry = \frac{1}{3} - 16x}\)

• TRANSLATE to standard form \(\mathrm{Ax + By = C}\):

First equation: Move all terms with variables to left side

\(\mathrm{48x - 72y = 30y + 24}\)

\(\mathrm{48x - 72y - 30y = 24}\)

\(\mathrm{48x - 102y = 24}\)

Second equation: Move all terms to get standard form

\(\mathrm{ry = \frac{1}{3} - 16x}\)

\(\mathrm{16x + ry = \frac{1}{3}}\)

2. INFER the condition for no solution

• A system has no solution when lines are parallel but distinct
• Lines are parallel when their coefficients are proportional: \(\mathrm{\frac{A_1}{A_2} = \frac{B_1}{B_2}}\)

3. SIMPLIFY to find the required ratio

• Compare x coefficients: \(\mathrm{\frac{48}{16} = 3}\)
• For parallel lines, y coefficients must have same ratio: \(\mathrm{\frac{-102}{r} = 3}\)

4. SIMPLIFY to solve for r

• Cross multiply: \(\mathrm{-102 = 3r}\)
• Divide by 3: \(\mathrm{r = \frac{-102}{3} = -34}\)

Answer: -34

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert the first equation to standard form, particularly combining \(\mathrm{-72y - 30y}\) incorrectly.

Many students write \(\mathrm{-72y - 30y = -42y}\) instead of \(\mathrm{-102y}\), leading to the wrong coefficient. This changes their proportionality setup to \(\mathrm{\frac{-42}{r} = 3}\), giving \(\mathrm{r = -14}\) instead of the correct \(\mathrm{r = -34}\).

Second Most Common Error:

Missing conceptual knowledge about parallel lines: Students may not remember that parallel lines require proportional coefficients, or they might confuse the conditions for no solution vs infinitely many solutions.

Without this key concept, they attempt to solve the system directly rather than analyzing the coefficient relationships, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students can connect algebraic manipulation skills with the geometric concept of parallel lines. The key insight is recognizing that "no solution" translates to a specific mathematical relationship between coefficients.