In the xy-plane, line k intersects the y-axis at the point \((0, -6)\) and passes through the point \((2, 2)\)....

1. TRANSLATE the problem information

• Given information:
  • Line k intersects y-axis at \((0, -6)\)
  • Line k passes through \((2, 2)\)
  • Point \((20, \mathrm{w})\) lies on line k
  • Need to find the value of w

• What this tells us: We have two known points on the line and need to find the y-coordinate of a third point.

2. INFER the solution approach

• To find w, we need the equation of line k first
• Since we have two points on the line, we can find the slope and y-intercept
• The y-intercept is already given: \(\mathrm{b} = -6\) (from point \((0, -6)\))

3. SIMPLIFY to find the slope

• Using slope formula with points \((0, -6)\) and \((2, 2)\):
• \(\mathrm{m} = \frac{2 - (-6)}{2 - 0}\)
  \(\mathrm{m} = \frac{8}{2}\)
  \(\mathrm{m} = 4\)

4. INFER the line equation

• Using \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\) form with \(\mathrm{m} = 4\) and \(\mathrm{b} = -6\):
• \(\mathrm{y} = 4\mathrm{x} - 6\)

5. SIMPLIFY to find w

• Substitute \(\mathrm{x} = 20\) into the equation:
• \(\mathrm{y} = 4(20) - 6\)
  \(\mathrm{y} = 80 - 6\)
  \(\mathrm{y} = 74\)
• Since w is the y-coordinate when \(\mathrm{x} = 20\), \(\mathrm{w} = 74\)

Answer: 74

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the coordinates or misunderstand what w represents.

Some students might think w is the x-coordinate instead of the y-coordinate, or they might mix up which point is which when setting up the slope calculation. This confusion about the coordinate system and what they're solving for can lead to completely incorrect setup and random guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating the slope or performing the final substitution.

For example, they might calculate the slope as \(\frac{2-(-6)}{2-0} = \frac{8}{2} = 4\) correctly, but then make an error like \(\mathrm{y} = 4(20) - 6 = 80 - 6 = 76\) instead of 74. These calculation mistakes are especially common when working with negative numbers in the y-intercept.

The Bottom Line:

This problem requires careful coordinate tracking and systematic equation-building. Success depends on clearly organizing the given points and methodically working through the slope-intercept form to build the line equation before finding the missing coordinate.