Three points lie on the same straight line. The coordinates of these points are \((1, -1)\), \((3, 5)\), and \((7,...

1. TRANSLATE the problem information

• Given information:
  • Three collinear points: \((1, -1)\), \((3, 5)\), and \((7, 17)\)
  • Need to find y-coordinate when \(\mathrm{x = 4}\)
• What this tells us: We need to find the equation of the line passing through these points, then use it to find the y-value.

2. INFER the approach

• Since all three points lie on the same line, we can use any two points to find the slope
• Once we have the slope, we can use point-slope form to get the line equation
• Strategy: Find slope → Write equation → Substitute \(\mathrm{x = 4}\)

3. SIMPLIFY to find the slope

Using points \((1, -1)\) and \((3, 5)\):

• Slope = \(\mathrm{\frac{y_2 - y_1}{x_2 - x_1}}\) = \(\mathrm{\frac{5 - (-1)}{3 - 1}}\) = \(\mathrm{\frac{6}{2}}\) = \(\mathrm{3}\)

4. SIMPLIFY to write the line equation

Using point-slope form with \((1, -1)\):

• \(\mathrm{y - (-1) = 3(x - 1)}\)
• \(\mathrm{y + 1 = 3x - 3}\)
• \(\mathrm{y = 3x - 4}\)

5. INFER verification step

Let's check our equation with the third point \((7, 17)\):

• \(\mathrm{y = 3(7) - 4 = 21 - 4 = 17}\) ✓

6. SIMPLIFY to find the final answer

When \(\mathrm{x = 4}\):

• \(\mathrm{y = 3(4) - 4 = 12 - 4 = 8}\)

Answer: C. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that they need to find the equation of the line first, instead trying to find patterns in the coordinates directly or attempting to use the distance formula.

This leads to confusion about how to approach the problem systematically, causing students to abandon a methodical solution and guess randomly among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when converting from point-slope form to slope-intercept form, such as:

• \(\mathrm{y + 1 = 3x - 3}\) becoming \(\mathrm{y = 3x - 2}\) (forgetting to subtract 1 from both sides)
• Or making arithmetic mistakes when substituting \(\mathrm{x = 4}\)

These errors typically result in answers like 10 or 6, which aren't among the choices, leading students to second-guess their approach and potentially select Choice D (11) as it seems "close" to their incorrect calculation.

The Bottom Line:

This problem tests whether students can systematically work with linear equations rather than trying shortcuts. The key insight is recognizing that finding the line equation is the most reliable path to the answer.