Straight lines — y = mx + c
Maths · GCSE · The School
Two numbers describe every line
In y = mx + c, m is the GRADIENT (how much y climbs for each step right) and c is the Y-INTERCEPT (where the line cuts the y-axis). y = 2x − 3 climbs 2 per step and cuts the axis at (0, −3). Read both before you draw anything.
Gradient from two points
Between (1, 3) and (4, 12): the rise is 12 − 3 = 9, the run is 4 − 1 = 3, so m = 9/3 = 3. Then feed one point into y = 3x + c to find c: 3 = 3(1) + c gives c = 0, so y = 3x. Rise over run, then solve for c.
Parallel and perpendicular
Parallel lines share the same m and never meet — y = 2x + 1 and y = 2x − 5 are a pair. Perpendicular lines have gradients that multiply to −1: a line with m = 2 meets a line with m = −1/2 at a right angle.
Negative gradients, and lines that break the pattern
Find the line through (−2, 7) and (2, −1). Rise is −1 − 7 = −8; run is 2 − (−2) = 4; so m = −8/4 = −2. A negative gradient means the line falls as you go right, and getting the sign wrong is the single most common error here — subtract the coordinates in the SAME order on top and bottom, every time. Then find c: substitute (2, −1) into y = −2x + c, giving −1 = −4 + c, so c = 3 and the line is y = −2x + 3. Check with the OTHER point: −2(−2) + 3 = 7, which matches. Two lines refuse the y = mx + c form. A horizontal line is y = 4 (gradient 0). A vertical line is x = 4, whose gradient is undefined, because the run is zero and you cannot divide by it.
Finding where a line crosses each axis
The line crosses the y-axis where x = 0, which is just c — no calculation needed. It crosses the x-axis where y = 0, so solve 0 = mx + c for x. For y = 2x − 6, the y-intercept is (0, −6) and the x-intercept comes from 0 = 2x − 6, giving x = 3, so (3, 0). Sketching a straight line only needs these two points — no table of values required.
The midpoint of a line segment
The midpoint of two points is the average of their x-coordinates and the average of their y-coordinates — halfway along in each direction. Between (1, 3) and (7, 9), the midpoint is ((1+7)/2, (3+9)/2) = (4, 6). It works because a straight line changes at a constant rate, so the halfway point in x lines up with the halfway point in y.