The line through two points

Through two distinct points there is exactly one straight line. Here we find the line through $A(1, 1)$ and $B(3, 5)$.

First we find the slope, which measures how much $y$ increases while $x$ increases by $1$. It is the difference in $y$ divided by the difference in $x$.

Next we build the equation from the slope $2$ and the fact that the line passes through $A(1, 1)$. Writing the line as $y = 2x + b$ and substituting $A$ gives $1 = 2 \cdot 1 + b$, so $b = -1$. The line is therefore $y = 2x - 1$.

Let us check with the other point $B(3, 5)$: $2 \cdot 3 - 1 = 5$, so the line does pass through $B$.

In general, the line through two points $(x_1, y_1)$ and $(x_2, y_2)$ has slope $\dfrac{y_2 - y_1}{x_2 - x_1}$. Calling this $m$, the line through $(x_1, y_1)$ with slope $m$ can be written as follows.

This routine — get the slope from two points, then build the equation through one of them — is the foundation for the intersection problems too. The large dots on the graph are the two points $A$ and $B$.