Geometry
Equation of a Straight Line – y = mx + c
Every straight (non-vertical) line on the coordinate plane can be described by a single equation: y = mx + c. Here, m is the gradient (how steep the line is), and c is the y-intercept (where it crosses the y-axis). Once you know m and c, you know absolutely everything about that line – you could redraw it perfectly without ever seeing the original graph.
This equation is one of the most useful tools in all of mathematics precisely because it connects an algebraic expression directly to a geometric picture – exactly the fusion Descartes and Fermat pioneered when they founded coordinate geometry in the 1630s.
Slope as Rise Over Run
Before treating m as an abstract number, see what it actually measures on the line itself. Pick any two points on the line and draw a right triangle between them: the vertical leg is how far you go up (the “rise”), and the horizontal leg is how far you go across (the “run”).
Here the rise is 6 and the run is 3, so the gradient (slope) is rise ÷ run = 6 ÷ 3 = 2 — and sure enough, this line is y = 2x. Whichever two points you pick on a straight line, rise ÷ run always gives the same number, because the line never curves — that constant steepness is exactly what m measures in y = mx + c.
How m and c Change the Line
Changing m rotates the line (makes it steeper or shallower); changing c slides the whole line up or down without changing its steepness. Three lines on the same grid make this easy to see:
y = x (grey) and y = 2x (red) both pass through the origin, but y = 2x is steeper because its m is bigger — that's what a larger m does. y = x (grey) and y = x + 3 (green) have the exact same steepness (both m = 1) but y = x + 3 is shifted 3 units higher, because its c is 3 instead of 0 — that's what changing c does. Every straight line's shape is controlled by exactly these two numbers.
Reading y = mx + c
In y = mx + c: m is the gradient (steepness/direction), and c is the y-intercept (where the line crosses the y-axis).
Gradient m = −2. y-intercept c = 7.
Substitute x = 3: y = 2(3) − 5 = 1. This matches the given y-value, so the point does lie on the line.
Real-Life Application
- Pricing plans: a fixed fee plus a per-unit rate is a straight-line equation.
- Travel graphs: constant-speed journeys are modelled by straight lines.
- Conversion formulas: many unit conversions (e.g. temperature) follow y = mx + c.
Key Takeaways
- y = mx + c describes every non-vertical straight line.
- m is the gradient; c is the y-intercept.
- A point lies on the line only if its coordinates satisfy the equation exactly.