The table above shows some pairs of x values and y values. Which of the following equations could represent the...

1. TRANSLATE the problem information

• Given information:
  • Table showing x and y value pairs: \((1,5), (2,7), (3,9), (4,11)\)
  • Four linear equation options in \(\mathrm{y = mx + b}\) form
  • Need to find which equation matches the data
• What this tells us: We need to find the equation that produces the exact y-values shown when we input the given x-values

2. INFER the testing strategy

• Key insight: The correct equation must work for ALL data points, not just one
• Strategy: Systematically substitute each x-value from the table into each equation and check if we get the corresponding y-value
• We can eliminate options as soon as we find one data point that doesn't work

3. SIMPLIFY by testing each equation

Testing Option A: y = 2x + 3

• When \(\mathrm{x = 1}\): \(\mathrm{y = 2(1) + 3 = 5}\) ✓ (matches table)
• When \(\mathrm{x = 2}\): \(\mathrm{y = 2(2) + 3 = 7}\) ✓ (matches table)
• When \(\mathrm{x = 3}\): \(\mathrm{y = 2(3) + 3 = 9}\) ✓ (matches table)
• When \(\mathrm{x = 4}\): \(\mathrm{y = 2(4) + 3 = 11}\) ✓ (matches table)

Testing Option B: y = 3x - 2

• When \(\mathrm{x = 1}\): \(\mathrm{y = 3(1) - 2 = 1 ≠ 5}\) ✗ (doesn't match)

Since Option B fails the first test, we can eliminate it.

Testing Option C: y = 4x - 1

• When \(\mathrm{x = 1}\): \(\mathrm{y = 4(1) - 1 = 3 ≠ 5}\) ✗ (doesn't match)

Option C also fails, so we eliminate it.

Testing Option D: y = 5x

• When \(\mathrm{x = 1}\): \(\mathrm{y = 5(1) = 5}\) ✓ (matches table)
• When \(\mathrm{x = 2}\): \(\mathrm{y = 5(2) = 10 ≠ 7}\) ✗ (doesn't match)

Option D fails on the second data point.

4. APPLY CONSTRAINTS to confirm the answer

Only Option A works for all four data points from the table.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students test only one data point (usually the first one) and assume the equation works if that single point matches.

For example, testing only \(\mathrm{x = 1}\) in Option D gives \(\mathrm{y = 5(1) = 5}\), which matches the table. Students might select Choice D (y = 5x) without checking the other data points, not realizing that \(\mathrm{x = 2}\) gives \(\mathrm{y = 10}\), not the required \(\mathrm{y = 7}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when substituting values, particularly with negative numbers or multi-step calculations.

For instance, when testing Option B with \(\mathrm{x = 1}\), they might calculate \(\mathrm{y = 3(1) - 2 = 3 - 2 = 5}\) (incorrect arithmetic: should be 1), leading them to think Option B works. This calculation error may lead them to select Choice B (y = 3x - 2).

The Bottom Line:

This problem requires methodical checking of all data points against each equation. Students who rush through the substitution process or don't verify all given points often select incorrect answers that work for only some of the data.