Undefined slope
Undefined slope is so often misunderstood and many students find it confusing that I thought it would be a good thing to dedicate a special lesson to explain it.
Basically, a slope that is undefined looks like the lines on the coordinate system you see below:
When a slope is undefined all you do is moving straight up or straight down only. You are not moving horizontally at all. In other words, the run is zero. The slope is therefore at its steepest.
A good real life example of undefined slope is an elevator since an elevator can only move straight up or straight down.
It got its name "undefined" from the fact that it is impossible to divide by zero.
Recall that 6/2 equal 3 because 3 × 2 = 6
However, it is impossible to do 5/0 because there exist no number you can multiply 0 by to get 5. We say that this division is undefined.
Now, let us try to get the slope for a couple of the lines above, say the line in the middle and the other line on the left.
For the line in the middle, the points are (1, 3) and (1, 9)
Let x
_{1} = 1 , y
_{1} = 9 and x
_{2} = 1 and y
_{2} = 3
m = (y
_{1} − y
_{2})/((x
_{1} − x
_{2})
m = (9 − 3)/(1 − 1)
m = 6/0
Since there exist no number you can multiply 0 by to get 6, we say that the slope is undefined.
For the line on the left, the points are (3, 4) and (3, 1)
Let x
_{1} = 3 , y
_{1} = 4 and x
_{2} = 3 and y
_{2} = 1
m = (y
_{1} − y
_{2})/((x
_{1} − x
_{2})
m = (4 − 1)/(3 − 3)
m = 3/0
Since there exist no number you can multiply 0 by to get 3, we say that the slope is undefined.
Notice that for the line in the middle, the xvalues are the same for both points (x
_{1} = x
_{2} = 1) . This is also the case for all the lines above.
In general, when the xvalues or xcoordinates are the same for both points, the slope is undefined. If you can make this observation, there is no need to compute the slope.
Take the undefined slope quiz below to see if you really understood this lesson.

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