2x + 3y = 710x + 15y = 35For each real number r, which of the following points lies on...

1. INFER the relationship between equations

• Given system:
  • \(\mathrm{2x + 3y = 7}\)
  • \(\mathrm{10x + 15y = 35}\)
• Key insight: Check if these equations are related by comparing coefficients
• Notice: \(\mathrm{10 = 5(2)}\), \(\mathrm{15 = 5(3)}\), and \(\mathrm{35 = 5(7)}\)
• This means: \(\mathrm{10x + 15y = 35}\) is exactly 5 times the first equation
• Conclusion: These are equivalent equations representing the same line!

2. INFER the solution strategy

• Since both equations represent the same line, any point satisfying one equation automatically satisfies both
• We only need to test which choice satisfies \(\mathrm{2x + 3y = 7}\) for any value of r
• The correct answer will make this equation true regardless of what r equals

3. SIMPLIFY by testing Choice B

• Choice B gives us the point: \(\mathrm{\left(-\frac{3r}{2} + \frac{7}{2}, r\right)}\)
• Substitute \(\mathrm{x = -\frac{3r}{2} + \frac{7}{2}}\) and \(\mathrm{y = r}\) into \(\mathrm{2x + 3y = 7}\):

\(\mathrm{2\left(-\frac{3r}{2} + \frac{7}{2}\right) + 3(r) = 7}\)

• Distribute the 2:

\(\mathrm{2\left(-\frac{3r}{2}\right) + 2\left(\frac{7}{2}\right) + 3r = 7}\)
\(\mathrm{-3r + 7 + 3r = 7}\)

• Combine like terms:

\(\mathrm{(-3r + 3r) + 7 = 7}\)
\(\mathrm{0 + 7 = 7}\)
\(\mathrm{7 = 7}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the two equations are equivalent and try to solve the system as if they were independent equations with a unique solution.

They might attempt to use elimination or substitution expecting to find specific values for x and y, rather than understanding this represents the same line with infinitely many solutions. This leads to confusion about what the question is asking and causes them to guess randomly among the choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt the correct approach but make algebraic errors when substituting the parametric expressions.

For example, when testing Choice B, they might incorrectly distribute: \(\mathrm{2\left(-\frac{3r}{2} + \frac{7}{2}\right) = -3r + \frac{7}{2}}\) instead of \(\mathrm{-3r + 7}\), or fail to recognize that \(\mathrm{-3r + 3r = 0}\). These calculation errors prevent them from seeing that the equation simplifies to \(\mathrm{7 = 7}\), leading them to incorrectly reject the right answer and potentially select Choice A, C, or D.

The Bottom Line:

This problem tests whether students understand that equivalent linear equations represent the same line, and whether they can work accurately with parametric expressions involving fractions and variables.