7x + 6y = 5 28x + 24y = 20 For each real number r, which of the following points...

1. INFER the relationship between equations

• Given system:
  • \(\mathrm{7x + 6y = 5}\)
  • \(\mathrm{28x + 24y = 20}\)
• Key insight: Check if these equations are equivalent by seeing if one is a multiple of the other

2. SIMPLIFY to verify equivalence

• Divide the second equation by 4:
  \(\mathrm{(28x + 24y)/4 = 20/4}\)
  \(\mathrm{7x + 6y = 5}\)
• This is identical to the first equation! The system represents the same line twice.

3. INFER what this means for the answer

• Since both equations represent the same line, any point that satisfies one equation automatically satisfies both
• We need to find which answer choice gives points on this line for any real number r

4. TRANSLATE the parametric approach

• To find points on the line \(\mathrm{7x + 6y = 5}\) using parameter r:
• Set \(\mathrm{y = r}\) (so the y-coordinate varies with r)
• Solve for x in terms of r

5. SIMPLIFY to find the x-coordinate

• Substitute \(\mathrm{y = r}\) into \(\mathrm{7x + 6y = 5}\):
  \(\mathrm{7x + 6r = 5}\)
  \(\mathrm{7x = 5 - 6r}\)
  \(\mathrm{x = (5 - 6r)/7 = -6r/7 + 5/7}\)
• Points on the line: \(\mathrm{(-6r/7 + 5/7, r)}\)

Answer: D (with noted discrepancy in fraction)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that the equations are equivalent and represent the same line.

Instead, they might try to solve the system as if the equations were independent, leading to confusion when they discover the system has infinitely many solutions. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "lies on the graph of each equation" means in the context of equivalent equations.

They might think they need different points for each equation, or they might not understand the parametric representation concept. This may lead them to select Choice A or Choice B by incorrectly setting up the parameter.

The Bottom Line:

The key insight is recognizing equivalent equations represent the same geometric object - one line. Once you see this, the problem becomes about finding a parametric representation rather than solving a traditional system.