Slope of line if Δx is equal to 0

Question

To calculate slope of line, I use $m = \frac{Δy}{Δx}$, but it doesn't work if $Δx = 0$. Is there a way to calculate slope of straight line? Is there anything I didn't notice?

Answer 1

The above formula for $m$ works for any line, except for $x=constant$. Suppose your line is described by $$y=mx+b$$ Take any two values of $x$ that are different. Then $$y_1=mx_1+b\\y_2=mx_2+b$$ You can subtract one from the other, and get $$y_1-y_2=m(x_1-x_2)$$If $x_1\ne x_2$ then $\Delta x=x_1-x_2$, $\Delta y=y_1-y_2$, and you get $$m=\frac{\Delta y}{\Delta x}$$

Note for $x=constant$, two different points will have $y_1\ne y_2$ and $x_1=x_2$. The same line can be described by $y_1>y_2$, which yields a slope of $m=\infty$, or $y_1< y_2$, which yields $m=-\infty$