Three points lie on the same line in the coordinate plane. The coordinates of these points are \((-2, 25)\), \((0,...

1. TRANSLATE the problem information

• Given information:
  • Three points on the same line: \((-2, 25), (0, 15), (1, 10)\)
  • Need to find the equation of this line

2. INFER the solution strategy

• To find a linear equation, I need slope (m) and y-intercept (b)
• I can use any two points to find slope with the slope formula
• Since I have the point \((0, 15)\), I can directly identify the y-intercept as 15

3. SIMPLIFY to find the slope

• Using points \((-2, 25)\) and \((0, 15)\):
  \(\mathrm{m} = \frac{15 - 25}{0 - (-2)}\)
  \(= \frac{-10}{2}\)
  \(= -5\)

4. INFER the complete equation

• With slope \(\mathrm{m} = -5\) and y-intercept \(\mathrm{b} = 15\):
  \(\mathrm{y} = -5\mathrm{x} + 15\)

5. SIMPLIFY to verify the equation works

• Check \((-2, 25)\):
  \(\mathrm{y} = -5(-2) + 15\)
  \(= 10 + 15\)
  \(= 25\) ✓
• Check \((0, 15)\):
  \(\mathrm{y} = -5(0) + 15\)
  \(= 0 + 15\)
  \(= 15\) ✓
• Check \((1, 10)\):
  \(\mathrm{y} = -5(1) + 15\)
  \(= -5 + 15\)
  \(= 10\) ✓

Answer: B) \(\mathrm{y} = -5\mathrm{x} + 15\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when calculating slope, especially with negative coordinates.

Students might calculate:
\(\mathrm{m} = \frac{15 - 25}{0 - (-2)}\)
\(= \frac{-10}{2}\)
\(= 5\) (forgetting the negative)
This leads them to think the equation is \(\mathrm{y} = 5\mathrm{x} + 15\), causing them to select Choice C (\(\mathrm{y} = 5\mathrm{x} + 10\)) after also making an error with the y-intercept.

Second Most Common Error:

Poor INFER reasoning: Not recognizing that the point \((0, 15)\) directly gives the y-intercept.

Students might use a more complicated approach, trying to substitute points into the general form and making algebraic errors in the process. This leads to confusion and potentially selecting Choice A (\(\mathrm{y} = -5\mathrm{x} + 10\)) with the correct slope but wrong y-intercept.

The Bottom Line:

This problem rewards students who recognize the strategic advantage of having a point where \(\mathrm{x} = 0\), allowing them to immediately identify the y-intercept rather than getting bogged down in unnecessary algebra.