Kaylani used fabric measuring 5 yards in length to make each suit for a men's choir. The relationship between the...

1. TRANSLATE the problem information

• Given information:
  • 5 yards of fabric needed per suit
  • \(\mathrm{x}\) = number of suits made
  • \(\mathrm{y}\) = total fabric purchased (in yards)
  • Equation: \(\mathrm{y - 5x = 6}\)

• What this tells us: We need to understand what each part of this equation means in the real-world context.

2. INFER the approach

• The key insight is that rearranging the equation will reveal the relationship between purchased and used fabric
• We should isolate y to see the equation in slope-intercept form

3. SIMPLIFY by rearranging the equation

• Starting with: \(\mathrm{y - 5x = 6}\)
• Add 5x to both sides: \(\mathrm{y = 5x + 6}\)

4. TRANSLATE each component back to the context

• Now we can see clearly:
  • \(\mathrm{y}\) = total fabric purchased
  • \(\mathrm{5x}\) = total fabric used (5 yards per suit × x suits)
  • \(\mathrm{6}\) = the difference between purchased and used

• This means: Total purchased = Total used + 6 additional yards

Answer: D. Kaylani purchased 6 yards more fabric than she used to make the suits.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to connect the abstract equation back to its real-world meaning. They might correctly rearrange the equation to \(\mathrm{y = 5x + 6}\) but then fail to interpret what the constant 6 represents in context. Instead, they might think 6 refers to a literal quantity like "6 suits" or "6 total yards" because these seem like simpler interpretations.

This may lead them to select Choice A (6 suits) or Choice B (6 total yards).

Second Most Common Error:

Insufficient INFER reasoning: Some students don't recognize that rearranging the equation is necessary to see the relationship clearly. They try to interpret \(\mathrm{y - 5x = 6}\) directly without rearranging, leading to confusion about what 6 represents when it's on the "opposite side" from y and 5x.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can move fluidly between algebraic representation and real-world context. The key challenge is recognizing that the constant term in a linear equation represents the y-intercept or, in this case, the "extra amount" beyond the variable relationship.