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CCSS.Math: , , ,

find the slope of the line that goes through the ordered pairs 7 comma negative 1 and negative 3 comma negative 1 let me just do a quick graph of these just so we can visualize what they look like so let me let me draw a quick graph over here so our first point is 7 comma negative 1 so 1 2 3 4 5 6 7 this is the x-axis 7 comma negative 1 so 7 comma negative 1 is right over there 7 negative 1 this of course is the y-axis and then the next point is negative 3 comma negative 1 so we go back 3 in the horizontal Direction negative 3 but the y-coordinate is still negative 1 it's still negative 1 so the line that connects these two points will look like this it will look like that now they're asking us to find the slope of the line that goes through the ordered pairs find the slope of this line and just to give a little bit of intuition here slope is a measure of a lines inclination and the way that it's defined slope is defined as rise over run or change in Y over change in X or sometimes you'll see it defined as as the variable M and then they'll define change in Y as just being the second y-coordinate - the first y-coordinate and then the change in X as the second x-coordinate - the first x-coordinate these are all different variations in slope but hopefully you appreciate that these are measuring inclination if I rise a ton when I run a little bit if I move a little bit in the X direction and I rise a bunch then I have a very steep line I have a very steep upward sloping line if I don't change at all in the when I run a bit then I have a very low slope and that's actually what's happening here I'm going from you could either use this as the starting point or view this as the starting point but let's do this as a starting point so this negative 3 comma 1 if I go from negative 3 if I go from negative 3 comma negative 1 to 7 comma negative 1 I'm running a good bid I'm going from negative three my x value is negative three here and it goes all the way to seven so my change in X here my change in X here is ten to go from negative 3 to 7 I changed my x value by 10 but what's my change in Y well my Y value here is negative 1 and my Y value over here is still negative 1 so my change in Y is 0 my change in Y is going to be 0 right my Y value does not change no matter how much I change my x value so the slope here and the slope here is going to be what's when we run 10 when we run 10 what was our rise how much did we change in Y well we didn't rise at all we didn't go up or down so the slope here is 0 or another way to think about it is this line has no inclination it's a completely flat it's a completely horizontal line so this should make sense this is a 0 the slope here is 0 and just to make sure that this gels with all of these other formulas that you might know but I want to make it very clear these are all just telling you rise over run or change in Y over change in X a way to measure inclination but let's just apply them just so hopefully it all makes sense to you so we could also say slope is change in Y over change in X if we take this if we take this to be our start and if we take this to be our end point then we would call this over here X 1 and then this is over here this is y 1 and then we would call this X 2 and we would call this Y 2 if this is our start point and that is our end point and so the slope here the change in Y y2 minus y1 so it's negative 1 minus negative 1 minus negative 1 all of that over X 2 negative 3 minus X 1 minus 7 so the numerator negative 1 minus negative 1 that's the same thing as negative 1 plus 1 and our denominator is negative 3 minus 7 which is negative 10 so once again negative 1 plus 1 is 0 over negative 10 and this is still going to be zero and the only reason why we got a negative ten here and a positive ten there is because we swapped the starting and the ending point in this example right over here we took this as the start point and made this this coordinate over here is the end point over here we swap them around seven negative one was our start point and negative three negative one is our end point so if we start over here if we start over here our change in X is going to be negative ten but our change in Y is still going to be zero so regardless of how you do it the slope of this line is zero it's a horizontal line