📈Equation of a Straight Line

Every line a ruler can draw — steep or flat — fits into one short formula. Read it well and two letters will tell you what the whole line looks like. The letters $m$ and $b$ work exactly like the variables you trained in Variables & the Function Machine.

y = mx + b: every letter has a job

$y = mx + b$

• $m$ is the slope: how steeply the line climbs;
• $b$ is the intercept: where the line meets the y-axis — its starting point when $x = 0$;
• Plot every point $(x, y)$ that satisfies the formula and they line up, perfectly straight.

Take $y = 2x + 1$: when $x = 0$, $y = 1$, so the line sets off from $(0, 1)$; each time $x$ grows by 1, $y$ grows by 2.

Slope m: rise over run

Slope answers one question: for each step forward, how far do you climb?

$m = \frac{\Delta y}{\Delta x}$

The Greek letter $\Delta$ (delta) means "change": $\Delta y$ is how much y changed, $\Delta x$ is how much x changed. For example, a line passing through $(1, 3)$ and $(4, 9)$:

$m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$

One step right, two steps up. The bigger $m$ is, the steeper the line: with $m > 0$ the line climbs to the right, with $m < 0$ it slides downhill.

Horizontal and vertical lines

Two lines deserve a closer look:

• Horizontal lines, like $y = 3$: x moves, y never budges, so the slope is $m = 0$ — no tilt at all;
• Vertical lines, like $x = 2$: y moves, x never budges. This is the one line $y = mx + b$ cannot write — $\Delta x = 0$, and dividing by zero has no meaning, so a vertical line has no slope.

m and b in real life

A phone plan: 19 yuan per month, plus 3 yuan per GB of data. Let $x$ be the GB used and $y$ the bill:

$y = 3x + 19$

$m = 3$ is the rate of change: each extra GB costs 3 more yuan. $b = 19$ is the starting point: the monthly fee you pay even with zero data. At 10 GB, $y = 3 \times 10 + 19 = 49$ yuan.

Plenty of real situations share this shape: $b$ is the fixed base, $m$ is the pay-per-use part.

Line Lab

Slope m1Intercept b2

y = x + 2

y-intercept

(0, 2)

x-intercept

(-2, 0)

Drag the sliders for $m$ and $b$. Move only $b$ first and watch the whole line slide; then move only $m$ and watch it pivot around the intercept — flatter, steeper, uphill to downhill.

Check yourself

✅

Quick quiz

↺ Try again

1. 1. A line passes through (0, 5) and rises 2 for every 1 it moves right. What is its equation?

   y = 2x + 5y = 5x + 2y = 2x - 5

2. 2. Which line is steepest?

   y = 3x + 1y = 2x + 8y = x + 100

3. 3. The plan is 19 yuan per month plus 3 yuan per GB. What does 10 GB cost?

   49 yuan30 yuan19 yuan