The equation of a straight line

Reading a line like a barcode

• Every straight line has a unique "signature" — its equation.
• If you can read the equation, you can tell the gradient, the intercept 截距, and where the line crosses every axis — without drawing it.
• The signature is just three letters: $y = mx + c$.

The form $y = mx + c$

• $m$ is the gradient (how steep the line is).
• $c$ is the $y$-intercept (where the line crosses the $y$-axis — the point $(0, c)$).
• Together, they completely describe a straight line.

In $y = 3x - 1$: the gradient $m = 3$ and the intercept $c = -1$. The figure shows both.

Reading $m$ and $c$ from an equation

• When the equation is already in $y = mx + c$ form, just read off the values:
• $y = 5x + 2$ → $m = 5$, $c = 2$.
• $y = -\dfrac{1}{2}x + 4$ → $m = -\dfrac{1}{2}$, $c = 4$.
• $y = 3x$ → $m = 3$, $c = 0$ (the line passes through the origin).

For $y=mx+c$ the line crosses the $y$-axis at $c$, and the gradient $m$ is the rise divided by the run

Finding the equation from a point and gradient

• Know the gradient $m$ and one point? Substitute into $y = mx + c$ and solve for $c$.
• Example: gradient $3$ through $(1, 2)$:
• $2 = 3(1) + c \Rightarrow c = 2 - 3 = -1$.
• The equation is $y = 3x - 1$.

Rearranging to $y = mx + c$

• Exams love giving equations like $ax + by = c$ and asking for the gradient.
• Step 1: isolate $y$ — move $ax$ to the other side, then divide by $b$.
• Step 2: read off $m$ and the intercept.
• Example: $2x + 3y = 12 \Rightarrow 3y = -2x + 12 \Rightarrow y = -\dfrac{2}{3}x + 4$.
• Gradient $= -\dfrac{2}{3}$; $y$-intercept $= 4$.

The line of best fit 最佳拟合线

• In science experiments, data points rarely fall on a perfect line.
• Scientists draw a line of best fit through the scatter — and its equation $y = mx + c$ summarises the whole experiment in one sentence.
• The gradient tells you the rate (e.g. how fast temperature rises per minute); the intercept tells you the starting value.

You've got it