## Equations of lines

Consider the line with gradient 3 and y-intercept 2. This passes through the point A(0, 2). Let B(x, y) be any point on this line.

\begin{align}\text{Gradient of interval}\ AB &= \dfrac{rise}{run}\\ &= \dfrac{y − 2}{x − 0}\\ &= \dfrac{y − 2}{x}\end{align}

The gradient of the line is 3.

\begin{align}\text{So}              \dfrac{y − 2}{x}&= 3\\ \text{Rearranging}              y − 2 &= 3x\\ y &= 3x + 2\end{align}

So the coordinates of B(x, y) satisfies y = 3x + 2. This is called the equation of the line.

Conversely, suppose that B(x, y) satisfies the equation y = 3x + 2, then  = 3 and it passes though the point (0, 2) so the point lies on the line with gradient 3 and y-intercept 2.

We summarise this by saying that the equation of the line is y = 3x + 2.