In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the...

1. INFER the condition for infinitely many solutions

• Key insight: A system of linear equations has infinitely many solutions when one equation is equivalent to the other
• This means one equation must be a scalar multiple of the other equation
• Every coefficient and constant in one equation equals the corresponding term in the other equation multiplied by the same number

2. INFER the relationship between the equations

• Compare the coefficients in both equations:
  • First equation: \(\mathrm{2x + 7y = 9}\)
  • Second equation: \(\mathrm{8x + 28y = a}\)
• Check if there's a consistent multiplier:
  • x-coefficients: \(\mathrm{8 ÷ 2 = 4}\)
  • y-coefficients: \(\mathrm{28 ÷ 7 = 4}\)
• The second equation's left side is exactly 4 times the first equation's left side

3. SIMPLIFY to find the value of a

• Since the left sides have a \(\mathrm{4:1}\) ratio, the right sides must also have this same ratio
• Therefore: \(\mathrm{a = 9 × 4 = 36}\)

Answer: C. 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the condition for infinitely many solutions

Many students know that systems can have one solution, no solution, or infinitely many solutions, but they don't understand that infinitely many solutions occurs when equations are equivalent. They might try to solve the system using elimination or substitution, getting confused when they can't find unique values for x and y. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct multiplier but make arithmetic errors

Students correctly identify that the second equation should be 4 times the first, but then calculate \(\mathrm{a = 9 × 4}\) incorrectly, perhaps getting \(\mathrm{a = 32}\) or another wrong value. Since 32 isn't an answer choice, this causes them to get stuck and guess.

The Bottom Line:

This problem requires students to shift from the typical "solve for x and y" mindset to understanding the structural relationship between equations. The key insight is recognizing that infinitely many solutions means the equations represent the same line—they must be scalar multiples of each other.