The Equation of a Straight Line
Imagine you had to describe a straight line to a friend over the phone — no drawing allowed.
You could read out point after point after point… and never finish, because a line has
infinitely many. Yet it turns out you only ever need to say two numbers. Tell
your friend how steep the line is and where it crosses the up-axis, and they
can draw the exact same line you're looking at. Every straight line on Earth is captured by
those two facts.
We bottle them in one tidy equation:
y = mx + c
The number m is the gradient — how steeply the line
tilts — and c is the y-intercept — the height at
which it crosses the y-axis. Read those two numbers off and you know
the whole line. (The gradient is exactly the
gradient you have
already measured as rise over run.)
For a straight line written as y = mx + c:
- m is the gradient — the steepness of the line (the coefficient of x);
- c is the y-intercept — where the line cuts the y-axis (the constant term);
- a bigger m makes the line steeper;
- changing c slides the whole line straight up or down.
Drive the line yourself
Pull the two sliders. Watch m tilt the line and
c slide it up and down without changing its tilt. Notice that the
line always crosses the y-axis at exactly the height
c.
Worked example 1 — build the line, then use it
A line has gradient m = 2 and y-intercept
c = 3. Write its equation, then find y when
x = 4.
Slot the two numbers into y = mx + c:
y = 2x + 3.
That's the whole line. It climbs two units for every one across and crosses the axis at
(0, 3). To find y at any
x, just substitute. At x = 4:
y = 2(4) + 3 = 8 + 3 = 11.
Worked example 2 — from two points to the equation
Find the equation of the line through (1, 5) and
(3, 11).
Step 1 — the gradient. Rise over run:
m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3.
Step 2 — the intercept. We know y = 3x + c; we just
need c. Substitute a point that's on the line — say
(1, 5) — and solve:
5 = 3(1) + c \;\Rightarrow\; 5 = 3 + c \;\Rightarrow\; c = 2.
So the equation is y = 3x + 2. Check with the other point:
3(3) + 2 = 11 ✓. Either point works — that's a handy way to catch a
slip.
Worked example 3 — a real straight-line story
A taxi charges a fixed \pounds 3 just to get in, then
\pounds 2 for every mile you travel. Let
y be the total fare and x the number of
miles. The fare is
y = 2x + 3.
Look what the two numbers mean here. The gradient
m = 2 is the rate — pounds per mile — and the
intercept c = 3 is the starting cost before you've
gone anywhere, the fare at x = 0. A 5-mile trip costs
y = 2(5) + 3 = \pounds 13. Almost every "fixed charge plus a rate"
situation — phone plans, gym membership, filling a pool — is secretly a
y = mx + c line.
In y = mx + c the two letters have fixed jobs, and mixing them up is
the classic slip:
-
m is the coefficient of x
(the number stuck to the x) — the gradient.
c is the lonely constant with no
x — the intercept. In y = 7x + 4 the
gradient is 7, not 4.
-
Rearrange into y = mx + c form before you read
anything off. The line 2y = 4x + 6 looks like it has
gradient 4 — but it isn't in the right form yet. Divide every term
by 2:
y = 2x + 3.
Now read it: the gradient is 2 and the intercept is
3. The y must be alone, with a coefficient
of exactly 1, first.
This little equation is the quiet workhorse behind an astonishing amount of grown-up maths. In
a science experiment you plot your measurements and draw the line of best fit —
the straight line that threads closest through a cloud of scattered points. That line has an
equation y = mx + c, and its m and
c get chosen — fitted — to match the data as well as possible.
Statisticians call this a regression
line, and it's the beating heart of the simplest machine-learning models: feed a
computer a pile of examples, let it fit the best m and
c, and now it can predict — tomorrow's temperature, a house
price, a trend. The same two-number recipe that draws a school graph is quietly behind economic
forecasts, trend lines, and the first models of artificial intelligence. Not bad for
y = mx + c.
See it explained