8x + 12y = 20 Which of the following equations is equivalent to the equation above?...

1. INFER the approach for finding equivalent equations

• Given equation: \(\mathrm{8x + 12y = 20}\)
• Key insight: Equivalent equations can be created by factoring out common factors from all terms
• Strategy: Find the greatest common divisor of all coefficients and the constant

2. INFER what numbers to examine for common factors

• Look at coefficients: 8, 12
• Look at constant term: 20
• Need to find the largest number that divides all three evenly

3. SIMPLIFY by finding the greatest common divisor

• \(\mathrm{8 = 2 × 4}\)
• \(\mathrm{12 = 3 × 4}\)
• \(\mathrm{20 = 5 × 4}\)
• Greatest common divisor = 4

4. SIMPLIFY by dividing each term by the common factor

• Original: \(\mathrm{8x + 12y = 20}\)
• Divide each term by 4:
  • \(\mathrm{8x ÷ 4 = 2x}\)
  • \(\mathrm{12y ÷ 4 = 3y}\)
  • \(\mathrm{20 ÷ 4 = 5}\)
• Equivalent equation: \(\mathrm{2x + 3y = 5}\)

Answer: A (\(\mathrm{2x + 3y = 5}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategy of factoring out common factors to create equivalent equations. Instead, they may try to solve for x and y individually or manipulate only parts of the equation.

This often leads them to get confused by the answer choices and resort to guessing, or they might select Choice D (\(\mathrm{2x + 12y = 20}\)) thinking they only need to change one coefficient.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand they need to find common factors but make arithmetic errors in the division step. For example, they might incorrectly divide 12y by 4 and get 4y instead of 3y.

This may lead them to select Choice C (\(\mathrm{8x + 3y = 5}\)) if they divided the wrong terms, or create confusion that results in guessing.

The Bottom Line:

This problem tests whether students understand that equivalent equations maintain the same solution set, and the most systematic way to find them is through factoring common divisors from all terms. The key insight is recognizing this as a factoring problem, not a solving problem.