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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:

slope-formula-image


Let's illustrate this with an example:

slope-formula-image


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:

slope-formula-image


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!


slope-formula-image


Draw the slope (in red)

slope-formula-image


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

slope-formula-image


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 y1 and call the other y2

Let y1 = 9

Let y2 = 1

Therefore, 9 − 1 = y1 − y2 = 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 x1 and call the other x2

Let x1 = 4

Let x2 = 2

Therefore, 4 − 2 = x1 − x2 = 2 = run

We can see then that

y1 − y2 = rise and

x1 − x2 = run

The formula becomes:

slope-formula-image


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 (x1,y1) = (8, 8) and (x2,y2) = (4, 4)

(y1 − y2) / (x1 − x2) = (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 (x1,y1) = (1, -5) and (x2,y2) = (2, -10)

(y1 − y2) / (x1 − x2) = (-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

(y2 − y1) / (x2 − x1)= (-10 − -5 )/(2 − 1) = (-10 + + 5)/1 = -5/1 = -5

In general slope = (y1 − y2) / (x1 − x2) = (y2 − y1) / (x2 − x1)

Now don't you wonder anymore about how to find the slope!






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