The sum of a number x and 7 is twice as large as a number y. The number y is...

1. TRANSLATE each statement into a mathematical equation

• First statement: "The sum of a number x and 7 is twice as large as a number y"
  • Sum of x and 7: \(\mathrm{x + 7}\)
  • "Twice as large as y" means: \(\mathrm{2y}\)
  • So: \(\mathrm{x + 7 = 2y}\)

• Second statement: "The number y is 3 less than the number x"
  • "y is 3 less than x" means: \(\mathrm{y = x - 3}\)

2. INFER which choice matches our translated system

• Our system is:
  • \(\mathrm{x + 7 = 2y}\)
  • \(\mathrm{y = x - 3}\)

• Looking at the choices, this exactly matches Choice A.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "y is 3 less than x" as "\(\mathrm{y = 3 - x}\)"

Students often think "3 less than x" means "start with 3 and subtract x," leading them to write \(\mathrm{y = 3 - x}\) instead of the correct \(\mathrm{y = x - 3}\). The correct interpretation is "start with x and subtract 3."

This may lead them to select Choice B (\(\mathrm{x + 7 = 2y}\); \(\mathrm{y = 3 - x}\)) or Choice D (\(\mathrm{2(x + 7) = y}\); \(\mathrm{y = 3 - x}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Misinterpreting "twice as large as y" as "twice the sum"

Students may think the phrase means "two times the entire sum (x + 7)," leading them to write \(\mathrm{2(x + 7) = y}\) instead of \(\mathrm{x + 7 = 2y}\). The correct interpretation is that the sum itself equals twice y.

This may lead them to select Choice C (\(\mathrm{2(x + 7) = y}\); \(\mathrm{y = x - 3}\)) or Choice D (\(\mathrm{2(x + 7) = y}\); \(\mathrm{y = 3 - x}\)).

The Bottom Line:

This problem tests precise translation skills. The key is methodically converting each English phrase into its correct mathematical equivalent, paying special attention to the order of terms in subtraction expressions.