Finding the equation of a line

Through (−2, 1) and (4, 4) there is exactly one line, and its equation is y = (1/2)x + 2.

Working backwards from the picture

So far you have started from an equation and drawn its line. This unit goes the other way: you know something about a line, such as two points on it, and you want its equation. Every case comes down to finding two numbers, the slope m and the y-intercept b, and writing y = mx + b. The easiest case is when both are given. A line with slope −2 and y-intercept (0, 4) has the equation y = −2x + 4, and there is nothing to calculate.

A slope and any point: point-slope form

Often the point you know is not on the y-axis. Suppose the slope is 2/3 and the line passes through (3, 1). Any other point (x, y) on the line must make the same slope with (3, 1), so (y − 1)/(x − 3) = 2/3. Multiply both sides by (x − 3) and you get y − 1 = (2/3)(x − 3). In general this is the point-slope form, y − y₁ = m(x − x₁), where (x₁, y₁) is the known point. To reach slope-intercept form, distribute and add: y − 1 = (2/3)x − 2, so y = (2/3)x − 1.

A shortcut gives the same result: put the point into y = mx + b and solve for b. With m = 2/3 and the point (3, 1): 1 = (2/3)(3) + b, so 1 = 2 + b and b = −1. Either way, check by substituting the known point into your final equation. (2/3) × 3 − 1 = 1, which matches the point. Watch the signs in point-slope form. For the point (−4, 2) the form reads y − 2 = m(x + 4), because x − (−4) is x + 4.

Two points

Given two points, find the slope first with the slope formula, then use either point as above. For (−2, 1) and (4, 4), m = (4 − 1)/(4 − (−2)) = 3/6 = 1/2. Using (−2, 1): 1 = (1/2)(−2) + b, so 1 = −1 + b and b = 2. The line is y = (1/2)x + 2. Check with the other point: (1/2)(4) + 2 = 4. It is good practice to use one point to find b and the other to check, because a slip in the slope would then show up as a mismatch.

Parallel and perpendicular lines use the same steps with a slope you borrow. The line parallel to y = 3x − 1 through (1, 5) has slope 3, so 5 = 3(1) + b and b = 2: y = 3x + 2. The line perpendicular to y = 2x + 3 through (4, 1) has slope −1/2, so 1 = (−1/2)(4) + b and b = 3: y = −(1/2)x + 3. In the graph lab, drag the two points anywhere on the grid and the panel writes out both forms of the line through them.