Every non-vertical line can be written in the form `y = mx + b` where `m` is the slope of the line and `b` is the value where the line crosses the `y`-axis. Given any two points, one can draw a line connecting them, and from these points we can determine the equation of this line. We will show how with two examples, first with a given pair of points and second, the general case for any points (with a word of caution).
Suppose we want to find the equation of the line connecting the points `(3,7)` and `(5,13)`. If these points are both on the line `y = mx + b`, then they satisfy the equation, i.e., `7 = 3m + b` and `13 = 5m + b`. But then we can solve for `b` in both and see that
```
7 - 3m = b = 13 - 5m
```
and so whatever our slope `m` is, it must satisfy `7 - 3m = 13 - 5m`. We can rearrange this as `13 - 7= 5m - 3m`, and so `6 = 2m`, or `m = 3`, and our line has a slope of 3. For completeness, we can plug this value in to either equation above to find the `y`-intercept `b`. Namely, `7 - 3*3 = 7 - 9 = -2` and so the line connecting the points `(3,7)` and `(5,13)` is given by the equation
In general, suppose we want to find the slope/equation of the line connecting two distinct points `(p,q)` and `(r,s)`. The exact same line of reasoning above works here: If these points lie on the line given by the equation `y = mx + b`, then we have
```
q = m*p + b
s = m*r + b
```
and so, solving for `b` gives
```
q - m*p = b = s - m*r.
```
We now solve for `m` as above, by rearranging to find
```
q - s = -m*r + m*p = m*(p - r).
```
Now (caution!) if `p - r` is not 0, we can divide both sides by this to get
```
m = (q - s)/(p - r)
```
and this is the slope of the line connecting the points `(p,q)` and `(r,s)`. But what if `p - r` is 0? We cannot divide by 0, and yet we can certainly draw a line between any two points, so what's going on? If `p - r = 0`, then `p = r`, so thinking geometrically about our points `(p,q)` and `(r,s)`, this means both points have the same `x` value. I.e., since our points are distinct, this means that the line connecting them is vertical, which cannot be described by a function `y = f(x)`, so this is why our computation to find a slope does not work, there is no number `m` so that both of our points satisfy the equation `y = mx + b`. (However, note that if we try to work with `x = ny + c`, we do get a 'slope' `n` and `x`-intercept `c` as a vertical line is a function *of `y`* despite not being a function of `x`.)
For completeness again, in the case of a non-vertical line where our slope was `(q - s)/(p - r)`, we may plug this into either of our equations above to solve for `b` and get the equation of the line, e.g.,
One easy way to remember how to compute the slope from two points is the saying "rise over run". Looking at the general case, the slope is the ratio `(q - s)/(p - r)`, where `q` and `s` are the `y` values of the points, and `p` and `r` are the `x` values, so the slope of the line is given by the difference in the `y` values - the *rise* - divided by the difference in the `x` values - the *run*.
