The equation of a straight line

The gradient–intercept form

• Every straight line can be written in gradient–intercept form:

$y = mx + c$

• $m$ — the gradient (steepness), the number multiplying $x$.
• $c$ — the $y$-intercept, the $y$-value where the line crosses the $y$-axis (where $x = 0$).
• Read off both numbers directly: in $y = 3x - 4$ the gradient is $3$ and the line crosses the $y$-axis at $-4$.

Finding the equation from a gradient and a point

• You need two facts: the gradient $m$, and one point $(x_1, y_1)$ the line passes through.
• The quickest route is the point–gradient form:

$y - y_1 = m(x - x_1)$

• Substitute $m$, $x_1$ and $y_1$, then rearrange into $y = mx + c$.
• If you are given two points instead, first find the gradient with $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, then use either point.

Rearranging into the general form

• NCEA answers are often wanted in the tidy general form $ax + by + c = 0$, with whole-number coefficients and no fractions.
• Multiply through by any denominator, then move every term to one side.