Essay

Documenting Path Alignment Formulas for Utility Infrastructure

As an apprentice field surveyor, you are drafting a technical report to document the layout of a straight fiber-optic cable line on a project grid. The cable runs along a linear trajectory directly between two key connection hubs located at coordinates (−2,−4)(-2, -4) and (1,−3)(1, -3). Your supervisor has requested that you write a clear, step-by-step mathematical explanation of how you manually find the path's equation in slope-intercept form (y=mx+by = mx + b).

In your report, write an essay outlining the complete 'two-points procedure' to find this line's equation. To satisfy your supervisor's documentation standards, you must explicitly:

  1. State the slope formula and show how to calculate the slope (mm) using the points (−2,−4)(-2, -4) and (1,−3)(1, -3), explaining the sign changes that occur when subtracting negative numbers.
  2. State the point-slope formula, y−y1=m(x−x1)y - y_1 = m(x - x_1), and show the initial equation after substituting your calculated slope and the point (1,−3)(1, -3).
  3. Detail the subsequent algebraic manipulations needed to simplify the equation, distribute the fractional slope, and subtract 3 from both sides using a common denominator to arrive at the final slope-intercept equation: y=13x−103y = \frac{1}{3}x - \frac{10}{3}.

Ensure your explanation is logical, sequential, and highlights how intermediate terms are handled.

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Updated 2026-09-19

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