A line through two points.

Step 1 of 4

The slope from two points

The board shows two white points, $(1, 1)$⁠ and $(3, 5)$⁠. Exactly one line passes through them. Its slope is the rise divided by the run between the points:

$m = \frac{y_2 - y_1}{x_2 - x_1}$$m =$$\frac{y_2 - y_1}{x_2 - x_1}$⁠

The run is the difference between the two $x$⁠ values, and the rise the difference between the two $y$⁠ values, in the same order.

Step 2 of 4

Move the line

Now move the line yourself. Slope m sets how steep it is, and Intercept n raises and lowers it without turning it. When the line passes through both points, the green triangle stretches exactly between them.

Step 3 of 4

Finding n

When you know the slope and one point on the line, find $n$⁠ by substituting: put the point’s $x$⁠ and $y$⁠ into $y = mx + n$⁠, and you are left with an equation in which only $n$⁠ is unknown. On the board Slope m is locked at $3$⁠, and the point is $(2, 4)$⁠.

Step 4 of 4

The equation of the line

Now all of it together: two points, a slope, $n$⁠, and an equation.