In the xy-plane, line k passes through the points \((0, -5)\) and \((1, -1)\). Which equation defines line k?

1. TRANSLATE the problem information

• Given information:
  • Line k passes through points \(\mathrm{(0, -5)}\) and \(\mathrm{(1, -1)}\)
  • Need to find equation in form \(\mathrm{y = mx + b}\)

2. INFER the strategic approach

• Since we have two points, we can find both the slope (m) and y-intercept (b)
• Key insight: Point \(\mathrm{(0, -5)}\) immediately tells us the y-intercept since \(\mathrm{x = 0}\)
• We'll use the slope formula with both points to find m

3. Extract the y-intercept

• From point \(\mathrm{(0, -5)}\): When \(\mathrm{x = 0}\), \(\mathrm{y = -5}\)
• Therefore: \(\mathrm{b = -5}\)

4. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute \(\mathrm{(0, -5)}\) as \(\mathrm{(x_1, y_1)}\) and \(\mathrm{(1, -1)}\) as \(\mathrm{(x_2, y_2)}\):
• \(\mathrm{m = \frac{-1 - (-5)}{1 - 0}}\)
• \(\mathrm{m = \frac{-1 + 5}{1}}\)
• \(\mathrm{m = \frac{4}{1} = 4}\)

5. Construct the final equation

• Substitute \(\mathrm{m = 4}\) and \(\mathrm{b = -5}\) into \(\mathrm{y = mx + b}\)
• \(\mathrm{y = 4x + (-5) = 4x - 5}\)

Answer: D. \(\mathrm{y = 4x - 5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when calculating the slope

Students often struggle with the double negative: \(\mathrm{(-1 - (-5))}\) and incorrectly compute this as \(\mathrm{(-1 - 5) = -6}\) instead of \(\mathrm{(-1 + 5) = 4}\). This gives them a slope of -6, leading to equation \(\mathrm{y = -6x - 5}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Mixing up which coordinate represents which variable in the slope formula

Some students incorrectly substitute the points, perhaps using \(\mathrm{m = \frac{1 - 0}{-1 - (-5)} = \frac{1}{4}}\), giving them slope \(\mathrm{\frac{1}{4}}\). Combined with the correct y-intercept of -5, this leads them to select Choice B (\(\mathrm{y = \frac{1}{4}x - 5}\)).

The Bottom Line:

This problem tests whether students can systematically extract information from coordinate pairs and avoid calculation errors with negative numbers. The key insight that point \(\mathrm{(0, -5)}\) immediately gives the y-intercept makes the problem more straightforward, but arithmetic precision with the slope calculation determines success.