How to find the slope
Here is how to find the slope. We saw in the lesson about
what is slope that slope is a measure of how steep a line is
That steepness can be measured with the following formula:
Let's illustrate this with an example:
For this situation, we see that the rise is 2 and the run is 4, so slope = 2/4
slope = 1/2 after simplification
what is the meaning of 1/2 ?
Since 1/2 is positive, you are going uphill. Now, suppose the unit is yard
1 is the rise. 2 is the run. This means that everytime you go up 1 yard, you go accross or horizontally 2 yards
This situation is not very steep. However, take a look at the following:
Here, the rise is 8 and the run or horizontal distance is 2
So, slope = 8/2 = 4 meters
4 meters = 4/1 meters. This means that each time you go 4 meters straight up, you only go 1 meter horizontally
This situation is very steep because you go up a lot compared to going horizontally
Now, let's see how to find the slope when we don't know the rise and the run.
If we graph the slope on the coordinate system, we will be able to derive another useful formula
Let us then try to put a slope of 8 as in previous example on the coordinate system.
Put a rise of 8 anywhere you wish. Then, put a run of 2. Here we go!
Draw the slope (in red)
If we remove everything in blue( rise and run), you are left with just the slope of the line
.
Then, label the two endpoints with their respective coordinates
The two coordinates (4, 9) and (2,1) can be used to get a slope of 4
Notice that 9 − 1 = 8. But 9 and 1 represent y-coordinates
Since we cannot call both coordinates y, we can call one y
_{1} and call the other y
_{2}
Let y
_{1} = 9
Let y
_{2} = 1
Therefore, 9 − 1 = y
_{1} − y
_{2} = 8 = rise
Notice also by the same token that 4 − 2 = 2. But 4 and 2 represent x-coordinates
Since we cannot call both coordinates x, we can call one x
_{1} and call the other x
_{2}
Let x
_{1} = 4
Let x
_{2} = 2
Therefore, 4 − 2 = x
_{1} − x
_{2} = 2 = run
We can see then that
y
_{1} − y
_{2} = rise and
x
_{1} − x
_{2} = run
The formula becomes:
So, if the rise and the run are not given, but you know at least two points, use the formula right above
Examples: How to find the slope when points are given
1) (8, 8) and (4, 4)
Let (x
_{1},y
_{1}) = (8, 8) and (x
_{2},y
_{2}) = (4, 4)
(y
_{1} − y
_{2}) / (x
_{1} − x
_{2}) = (8 − 4 )/(8 − 4 ) = 4/4 = 1
Since 1 is positive, the line goes up as you move from left to right
2) (1, -5) and (2, -10)
Let (x
_{1},y
_{1}) = (1, -5) and (x
_{2},y
_{2}) = (2, -10)
(y
_{1} − y
_{2}) / (x
_{1} − x
_{2}) = (-5 − -10 )/(1 − 2) = (-5 + + 10)/-1 = 5/-1 = -5
Since -5 is negative, the line goes down as you move from left to right
Notice that
(y
_{2} − y
_{1}) / (x
_{2} − x
_{1})= (-10 − -5 )/(2 − 1) = (-10 + + 5)/1 = -5/1 = -5
In general slope = (y
_{1} − y
_{2}) / (x
_{1} − x
_{2}) = (y
_{2} − y
_{1}) / (x
_{2} − x
_{1})
Now don't you wonder anymore about how to find the slope!