-{15x + 25y = 65} One of the two equations in a system of linear equations is given. The system...

1. INFER the key relationship

• Given: One equation in a system has infinitely many solutions
• Key insight: For infinitely many solutions, both equations must be equivalent (represent the same line)
• Strategy: Simplify the given equation and each answer choice to see which ones match

2. SIMPLIFY the given equation to lowest terms

• Given: \(-15x + 25y = 65\)
• Find GCD of coefficients: \(\mathrm{GCD}(15, 25, 65) = 5\)
• Divide entire equation by 5: \(-3x + 5y = 13\)

3. SIMPLIFY each answer choice and compare

Choice A: 12x + 20y = 52

• \(\mathrm{GCD}(12, 20, 52) = 4\)
• Divide by 4: \(3x + 5y = 13\)
• Compare to \(-3x + 5y = 13\) → NOT equivalent

Choice B: 12x + 20y = -52

• Divide by 4: \(3x + 5y = -13\)
• Compare to \(-3x + 5y = 13\) → NOT equivalent

Choice C: -12x + 20y = 52

• Divide by 4: \(-3x + 5y = 13\)
• Compare to \(-3x + 5y = 13\) → EQUIVALENT! ✓

Choice D: -12x + 20y = -52

• Divide by 4: \(-3x + 5y = -13\)
• Compare to \(-3x + 5y = 13\) → NOT equivalent

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students attempt to compare equations without reducing them to lowest terms, or make arithmetic errors when finding the GCD or performing division.

For example, they might incorrectly think \(-15x + 25y = 65\) is equivalent to \(-12x + 20y = 52\) just because the coefficients "look similar," without actually simplifying both equations. This leads to random guessing between the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students don't understand what "infinitely many solutions" means mathematically, so they don't realize they need to find equivalent equations.

They might think they need to solve a system or substitute values, completely missing that the equations must represent the same line. This may lead them to select Choice A (\(12x + 20y = 52\)) by incorrectly trying to add or manipulate the original equation.

The Bottom Line:

This problem tests whether students understand the geometric meaning of "infinitely many solutions" (same line = equivalent equations) and can execute the algebraic skill of reducing equations to compare them properly.