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| Worked Example Slope from Two Points |
Worked Example Slope from Two Points
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
y = 3x - 2
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.,
q = p*(q - s)/(p - r) + b
and so
b = q - p*(q - s)/(p - r) = [q*(p - r) - p*(q - s)]/(p - r) = (q*r - p*s)/(p - r)
and the (non-vertical) line connecting the points (p,q) and (r,s) is given by
y = x*(q - s)/(p - r) + (q*r - p*s)/(p - r).
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.