The table shows three values of x and their corresponding values of y, where s is a constant. There is...

1. TRANSLATE the problem information

• Given information:
  • Three coordinate pairs: \((-2s, 24), (-s, 21), (s, 15)\)
  • Linear relationship between x and y
  • Need to find which equation represents this relationship

2. INFER the approach

• Since we have a linear relationship, we need to find the equation of the line
• Strategy: Use two points to find the slope, then use point-slope form
• Choose two points that make calculations easier: \((-s, 21)\) and \((s, 15)\)

3. Calculate the slope

• Using slope formula: \(m = \frac{y_2-y_1}{x_2-x_1}\)
• \(m = \frac{15 - 21}{s - (-s)}\)
  \(m = \frac{-6}{2s}\)
  \(m = \frac{-3}{s}\)

4. INFER which point-slope approach to use

• Use point-slope form: \(y - y_1 = m(x - x_1)\)
• Using point \((s, 15)\) and slope \(\frac{-3}{s}\):
• \(y - 15 = \left(\frac{-3}{s}\right)(x - s)\)

5. SIMPLIFY to standard form

• Distribute: \(y - 15 = \frac{-3x}{s} + 3\)
• Add 15: \(y = \frac{-3x}{s} + 18\)
• Multiply both sides by s: \(sy = -3x + 18s\)
• Rearrange: \(3x + sy = 18s\)

6. Verify with another point

• Check \((-2s, 24)\): \(3(-2s) + s(24) = -6s + 24s = 18s\) ✓

Answer: B. \(3x + sy = 18s\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to substitute the given points directly into the answer choices without first deriving the general linear equation. They might plug in values hoping to eliminate wrong answers, but this approach becomes confusing when dealing with the parameter s in multiple places.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the slope and set up point-slope form, but make algebraic errors when converting to standard form. Common mistakes include sign errors when distributing or incorrectly multiplying through by s.

This may lead them to select Choice A (\(sx + 3y = 18s\)) where they've swapped coefficients, or Choice C (\(3x + sy = 18\)) where they've lost the s in the constant term.

The Bottom Line:

This problem tests both strategic thinking (knowing to find slope first) and careful algebraic manipulation. Students need to resist the temptation to work backwards from answer choices and instead build the equation systematically from the given data.