How to find the slopeHere 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 ycoordinates 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 xcoordinates 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! 




