Writing a linear equation means constructing the equation of a line from given information: a slope and a point, two points, a graph, a table, or a verbal description. The process varies slightly depending on the input: given slope and y-intercept, use y = mx + b directly; given slope and any point, use point-slope form; given two points, compute slope first. This skill is the inverse of graphing — instead of going from equation to picture, you go from picture (or data) to equation. It is essential for modeling real-world linear relationships.
Practice with all input types: given m and b, given m and a point, given two points, given a graph, given a word problem. For each, decide which form to write in and convert if needed. Emphasize word problems where students must identify the slope (rate of change) and a known point or initial value from context. Compare the resulting equations to verify they describe the same line.
You already know what slope means and how to read and graph equations in slope-intercept or point-slope form. Now you are doing the inverse: *given* information about a line — a slope and a point, two points, a graph, a table, or a verbal description — *construct* the equation. This is the skill of mathematical modeling in miniature: translating a real-world situation into a formula that can make predictions.
The key insight is that a line is completely determined by two pieces of information. When the given information directly includes the slope and the y-intercept, use slope-intercept form y = mx + b by reading off m and b immediately. When you have the slope and any other specific point (not necessarily the y-intercept), use point-slope form y − y₁ = m(x − x₁), plugging in the known values directly — no need to hunt for b first. Given two points (x₁, y₁) and (x₂, y₂), compute slope first using m = (y₂ − y₁)/(x₂ − x₁), then feed that slope and either point into point-slope form. In all cases, the goal is the same equation; only the path to get there differs.
Word problems require one more translation step. Look for the rate of change (slope) hidden in phrases like "per hour," "each day," or "for every additional unit." Look for the initial value (y-intercept) in phrases like "starts at," "begins with," or the value when the independent variable equals zero. For example: "A plumber charges $75 per hour plus a $50 service fee." Here m = 75 (dollars per hour) and b = 50 (the flat fee at zero hours), giving cost = 75h + 50. The equation lets you predict cost for any number of hours without recomputing from scratch every time.
A common error is reaching for slope-intercept form even when the y-intercept is not given, which forces an unnecessary extra step. If you are given slope and a non-y-intercept point, write point-slope form first: y − y₁ = m(x − x₁). Simplify to slope-intercept form afterward if needed. Another pitfall: computing slope as Δx/Δy (run over rise) instead of Δy/Δx (rise over run). Label your points explicitly — (x₁, y₁) and (x₂, y₂) — and write the slope formula before substituting to avoid this swap.