All right so right now I want to go over slope is very much related to our previous video and which was about coordinate geometry you know having different points on a coordinate system like what we had before you can make different shapes like squares triangles and so on and really what we focused on we`re making shapes from lines that we worked on had two points either we use the distance formula or the Pythagorean theorem to find the distance between those two points but there were two points nonetheless and these lines have slopes are the steepness so steepness of the line.
So a huge hill can be very steep a small hill won`t be that steep and it can go in any direction so let`s talk about slope so in this case we check what the line is doing from left to right so from left to right this line is increasing in Y it`s going up which means that it has a positive slope it has a steepness that is positive to give it as an upward hill how do you find the steepness or this slope well we represented by M and it`s rise over run so you`ll hear this over and over again this just means give me the distance that it rises between any two points and give me the distance that it runs between any two points and divide the two and that`s my slope so rise that means give me a difference in Y difference in Y and then the run is a difference in X well we know the differences in Y and X they correspond to different directions so a difference in Y would be the vertical direction a difference in X would be the horizontal direction this looks familiar then because before we decompose this line into x and y x and y-components horizontal and vertical components so let`s say this point here one comma one and let`s say this here is negative 1 comma negative 1 and now we can find out what this point here is and that point is 1 comma negative 1 all right so how do we find the slope the slope is the steepness so the rise would be the difference in Y so we can represent that by y2 minus y1 and the difference in X is x2 minus x1 we`re using the same notation that we had with the distance formula so let`s make this x1 y1 it doesn`t matter which point you use first and this x2 y2 so now what is y2 minus y1 that would be negative 1 minus 1 so negative 1 minus 1 and X 2 minus X 1 is negative 1 minus 1 which means that we have negative 2 over negative 2 or 1 so our slope is 1 is 1 positive or negative it`s positive which makes sense because in the beginning I said it`s going upward so a slope that is greater than 0 is an upward slope and/or positive slope a slope that is less than 0 is a negative or a downward slope and what if a slope is just 0 if M equals 0 then that`s just a horizontal slope which means you`re not you`re not rising anywhere so let`s look at a slope that has a slope that`s 0 what kind of line would have a slope that`s 0 well this line of course it`s not rising it`s just going to the left or to the right you`re basically walking on flat ground there`s no rise but there`s only run so M equals rise over run will rise as 0 and run is whatever you want it to be could be 5 it could be 10 but the point is M will be 0 0 divided by any number is 0 except 0 of course you can`t divide 0 by 0 which brings me to my next point you can have an undefined slope and the undefined slope will be in the case of a vertical line so here if you have this vertical line well is there any run no there`s no run but there`s only a rise so that means that M is going to equal some run on 0 run on the bottom some rise could be five well five you can`t divide by zero so this would be undefined slope so if you ever see a vertical line and they ask you to find the slope it`s undefined if they give you a very hard-on to line and that`s good to find the slope it is just zero and then the rest are in between.
The rest are in between zero and undefined you could have a negative 0.5 slopes one half slope ten slope and so on so that`s how you find slopes okay so that`s slope the steepness of a line alright so now let`s try a problem find the slope between these two points one zero and ten and negative five and tell me if it`s positive or negative
M equals well let`s label this x1 y1 x2 y2 well that`s y2 minus y1 which is negative five minus zero over x2 minus x1 which is ten minus one so that is a negative five over nine and that is a negative number so it`s a negative slope so it`s going to look will this not drawn to scale but it`s going to be a line that`s going down from left to right and now that we know about slope let`s talk about the y-intercept so I`ll write this here the y-intercept all of these are lines we have different properties of lines we have the distance we have the slope now we need the y-intercept so let me draw another coordinate plane and here I`ll draw a line so this is my line now you see where the line hits the y-axis that is the y-intercept it`s just it`s a point.
Or you can think of it as just a y-value either way it`s going to be where the line it`s the y-axis here`s the y-axis where does this red line hit the y-axis right here so let`s make up at that point would be let`s say that`s 0 comma 1 so that point is 0 comma 1 so the y intercept here is y equals 1 you can think of the point is this 0 comma 1 either way and now let`s find the slope of this line let`s say another point here would be let`s say 1 comma 3 this is 1 comma 3 so the slope would be rise over run so 3 minus 1 which is 2 over 1 minus 0 which is 1 slope of 2 so this has a y-intercept I read it y-intercept of 1 and a slope of 2 great now what I want to do and you`ll see why am I want to call y-intercepts not by its name y-intercept but by a variable and I want that variable to equal B so B stands for y-intercept M stands for slope as we know and now we can actually write the equation of a line a line is a function so it takes in an x-value and gives out an output which is a y-value and we can draw up you can write the equation of a line so that if you give it any x value it`ll give you the y value that corresponds to it in other words if I told you that X was 1 here and I plug that into the input of the line it would give me 3 as a y-value so y equals f of X and what does that function equally well in this case before I show you how to do it I`ll tell you that the equation of the line is going to be 2x plus 1.
Let`s see if it works first of all if the equation of the line is 2x plus 1 that means that f of 1 should be well that should be 3 let`s see 2 times 1 plus 1 yes that equals 3 now if I plug in zero we should get 1 because here zero we have one so two times zero plus one equals yes it equals one because if we go to the x coordinate of zero then we have to go up one to be on that point on the line so great it seems like this line equation works but how did I get it how did I go from just knowing two points how did I go from this knowing two points to finding the equation of a line well it turns out that all you need is two points so I`ll write this here all you need are two points to find the equation the equation of a line of a line so the form of the equation of a line the general form would be y equals MX plus B B is the y-intercept so y-intercept M is the slope and x and y are the two variables y is the output and X is the input so you can think of Y is also f of X if you want to so great y equals MX plus B that has a ring to it so you need to memorize that now let`s try and use this again so we get to practice with it find the equation of a line given that a line has these two points let`s say 3 commas 4 & 5 comma 6 so find the equation of a line given these two points well the equation of a line is y equals MX plus B if we have M and B then we found it let`s find m M is y2 minus y1 over x2 minus x1 we have the two points so we can definitely find that very easy to find the slope this equals 6 minus 4 over 5 minus 3 which is 2 over 2 so that equals 1 so M equals 1 you can just write it here M equals 1.
We have the slope now I want to find the y-intercept so the y intercept or B is going to be is going to be a quantity that we find once we plug in one of these points to in our equation so right now what do we have so right now we have our equation as y equals MX plus B but M is 1 so we have y equals x plus B ok so using this equation let`s plug in any point because all these points all of two of these points are on the line let`s plug in 5 6 so X is 5 y is 6 so we have 6 equals 5 plus B and great now we`re going to solve for B so we get here we get B equals 1 so we have B equals one and just like that we found em and we found B so again the rule is for em use the equation you know y2 minus y1 over x2 minus x1 for the y-intercept pick a point that you know is on the line and put it into our equation once you found the slope and then you can solve for B very easily so now that we have Y now that we have the slope and the y-intercept we can say our equation is y equals 1 X plus 1 or X plus 1 that is our equation y equals x plus 1 great and so we can graph that if we want to let`s graph it so our equation is y equals x plus 1 how do we graph this well this is interesting we need to look at the coordinates oh this is X and this is y so where is the y-intercept it`s 1 1 is here and so we have 1 here now from this y-intercept we know what the slope is the slope is 1 if the slope is 1 that means the rise over run is also equal to run is equal to 1 well we have rise over run equaling to 1 but it seems like once only on the numerator but 1 is the same as 1 over 1 so the rise is 1 and the run is 1 remember what rise and run means rise means to go up or down if it`s negative you go down if it`s positive if you go up and run means to go right or left if it`s positive you go right if it`s negative you go left in this case our rise is 1 and our run is 1 so that means we`re going to go up 1 and to the right 1 to make our line and we keep on using that rule so up 1 and well I`m not going to draw the point there so but we`ll have up 1 and then to the right 1 and that`s our new point up 1 and then to the right 1 that`s our new point up 1 and to the right 1 that`s our new point and we can go down 2 to continue the line down 1 and to the left 1 down 1 and to the left 1 and hey look we have a whole bunch of different points for our line so this is our line this line is y equals plus one and that`s awesome we just graphed the line just using the equation and we found the equation using our techniques to find the slope in the y-intercept so that is the graph y equals x plus 1 awesome let`s try another example of this so I`m going to give you two points and I`m going to draw it on the coordinate axis this time and so we have the point 3 comma 5 and negative 2 comma negative 4 so find the equation of the line and also oh yeah just find the equation of the line because if you do you already know you`re going to find the slope in the y-intercept great so sort of think of how you would do this the way I would approach it is first find the slope and then the 1 intercept so I`m going to label these points x1 y1 x2 y2 and once you do enough of these it`ll become second nature so given that I know the slope is y2 minus y1 which is 5 minus negative 4 which is 9 over x2 minus x1 which is 5 so slope is 9 over 5 great now how do we find the y-intercept well we know the equation is y equals MX plus B so we already have the slope so it`s y equals 9 over 5x plus B but we still need B which is the y-intercept so we can plug in any point let`s pick this point to plug in so we`re going to plug in the Y values and I get it for the x value as negative 2 so we have 9 over 5 times negative 2 plus B so let`s solve for B we have negative 4 T equals negative 18 over 5 plus B and B would equal B would equal negative 2 over 5 and you can just use your skills of solving for variables and linear equations to find that so B equals negative 2 over 5 all right so we have the equation of the line then we have y equals MX plus B or y equals 9 over 5x minus 2 over 5 and so 9 over 5 is a positive number which is why we have a slope that`s positive and here`s an interesting point so we have the equation but even though the y-intercept is negative 2 over 5 it doesn`t look like this line is hitting it there that`s because I drew this line before actually finding the y-intercept which now we know it`s not a good idea because I just drew it from just from my hand I just free drew it so if you free draw a line you`re about to get something wrong so the best way of doing this would be to draw the line afterwards so let`s do that now that we have the equation from the two points now we can draw the line and so we`ll get a better and accurate representation the line is still correct the equation is so correct but I just wanted to look like what it is so we start at negative 2 over 5 on the y-axis so that we can just imagine is there.
So then this equals y so then we have a slope of 9 over 5 so we`re gonna go up 9 across 5 so up 9 across 5 up 9 across 5 and you see how you can just estimate a down nine across negative five and then so on so our line would look like that so it`s the same thing as what we had before just shifted a bit down this was shifted a bit down but the same points are there the one point would be what did we have negative 2 negative 4 and then another point would be 3 commas 5.
Everything is still the same we just drew it better so when they give you two points remember the things to do to find the equation is number one slope find the slope number two find the y-intercept by plugging at a point so plug in plug in any points plug in any point that they give you to find B which is the y-intercept and then write out the line and then after that what you want to do is just graph remember graph the y-intercept we did that here we put that point and then do the rise of a run rise over run some of them out rise run some amount and then can do it down and then just connect the dots so that`s it`s pretty simple that`s how you find a line given two points and then you can give in you can be given some other information that doesn`t include two points and you can still find the line but the most straightforward way is to be given two points and you find the line the equation of the line that way alright great so now I want to talk about some properties of lines so we have lines with slopes okay lines with slopes and now these slopes can be any number what if we have two lines so line a and line B line B but line a has a slope of two and line B has a slope of two.
Does that mean that the that they`re the same line no it doesn`t because if they have the same slope they can be anyone on the rectangular grid all it means is that they have the same slope so I can draw a line a like this that looks a bit squiggly so how I draw a line a like this and label that a and then I can draw a line B like this you see they have the same steepness in the same slope which is M equals to M equals two but they`re different lines why are they different lines because they have different y-intercepts this y-intercept for a is here this y-intercept is here so same slope different y-intercepts but what do you notice about these two lines well they`re parallel what does it mean to be parallel it means that these lines have the same slope which and then turn means that they go in the same direction but never cross never cross they never intersect they go in the same direction so they`re both going up in the same direction and they never cross and likewise they`re both going down in the same direction and never cross so the rule is then if we have if two lines if two lines have the same slope in this case it`s M equals to the same slope then they are and this is important they are parallel and we just explained what that definition is so that`s the first part I want to go over if two lines have the same slope they are parallel so let me give you an example with that are these two lines parallel that I`m going to draw I am going to actually I`ll just give you the points we have line a that has two points that are 0 0 and 1 comma 1 and line B line B that has two points at our 3 comma 3 and 2 comma 2 ok let`s see what is the slope of the first line well this is x1 y1 x2 y2 and let`s find it so M equals y2 minus y1 so that`s 1 minus 0 and x2 minus x1 is one minus zero so slope of line a is 1 slope of line 2 would be 2 minus 3 is negative 1 and a 2 minus 3 is negative 1 which is 1 yes they have the same slope so they are parallel and then you can also test if lines are parallel when you`re given the line equations itself so let`s give ourselves to line equations y equals 2x plus 5 and y equals 2x minus 100 well what`s the slope with the first line the slope here is M equals 2 the slope here and the second line is M equals 2 so yes they are parallel so that`s how you tell by just when you`re given the equations of the line or the lines itself so we talked about parallel lines which should be pretty straightforward all you`re looking for is similar slopes or same slopes but lines are not always parallel they can be perpendicular as well and we`ve seen perpendicular lines before when we talked about the Pythagorean theorem this is a 90-degree angle this line a alcohol on this line B are perpendicular meaning that they intersect at a 90 degree angle so these lines intersect at 90-degree angle these lines also intersect at a 90-degree angle so these are called perpendicular lines and what you get into higher math or even maybe now they`ll also be called orthogonal lines but either way they both mean they meet at 90 degrees so you see here they definitely do not have the same slope because they`re intersecting and we know that they`re perpendicular so what is a relationship between their slopes well let`s play around with it let`s draw two lines that are perpendicular and see what we can conclude from them so I`m going to give you two lines the first line will be y is equal to 3x plus 2 and then the second line will be y equals to negative 1/3 X plus 1 ok so we can we can graph them now since I didn`t tell you how you can tell if they`re perpendicular you shouldn`t believe me when I say this when I just draw it in it looks perpendicular you should believe me when I give you proof or give you an equation so let`s just write out just so we see it first so let`s see and you can always derive this later but it`s not that important right now so 3x plus 2 has a y-intercept of 2.
If we have a y-intercept of 2 we can write it here as 2 and then it has a rise over run as 3 over 1 so I go up 3 over 1 up 3 over 1 and this is our line that`s the first line and now here we have a y-intercept of 1 and here slope is negative 1 over 3 which means we go down 1 and across 3 so down 1 across 3 and down 1 across 3 and so on so I`ll actually write that in blue I`ll draw it in blue and these lines definitely look perpendicular and they are I`m just telling you they are but what can we tell or what.
What`s the relationship between their slopes that tells us these lines are perpendicular well the first slope is M equals 3 the second slope is M equals negative 1/3 what`s the relationship between these two one is 3 1 is negative 1/3 well it turns out that these are what`s called negative their negatives of each other but they`re not only that they`re negative reciprocals negative reciprocals of each other and so reciprocals which means that if one is three we take the negative of three and then we flip it to reciprocate means to flip it so you go to negative three and then we flip it if you flip three or negative 3 you get negative 1/3.
Let`s just talk about reciprocals really quickly the reciprocal of 2 is 1/2 the reciprocal of 2/3 is 3 over 2 and so on so if we have an M of 1/2 then a slope of a line that is perpendicular will have a slope of negative 2 or at negative 2 over 1 you see that and so these two lines have slopes are negative reciprocals of each other 3 and negative 1/3 so you know that when you graph them they`re going to be perpendicular they`re going to have this right angle great so now let`s summarize we went over properties of lines we went over that lines can be parallel and they can be perpendicular ok great so these are the two main ways that we`re going to analyze lines parallel means the same slope that they have the same slope and you know how to find slope and perpendicular means slopes are our negative reciprocals of each other of each other ok so if one is 5 then the other one would be negative 1 over 5 so those are the two ways we`re analyzing lines and in our next video we`ll go over some problems all right let`s go over the problems for the slope topic ok so you need to figure out whether these two lines for a B and C are parallel or perpendicular ok so the a is y equals 3x minus 10 and y equals negative 1 over 3x plus 40 are they parallel or perpendicular next for B y equals 1 over 5x plus 3 + 5 y equals x plus 15 SI y equals 2 and x equals 3 so why don`t you pause take a few minutes and do these problems alright so for Part A well we can see that they both have different slopes but the slope for the first line is 3 the slope for the second one is a negative reciprocal of 3 or negative 1/3 so these guys are perpendicular these lines are perpendicular and that means that they intersect at a 90 degree angle let`s check B 1 sub is 1/5 and the other slope is a weight we need to put it in the form y equals MX plus B so the form is y equals MX plus B so to get that we need to solve for y so I`ll divide by 5/5 so that we get y equals 1/5 X because X divided by 5 is 1/5 X plus 15 divided by 5 is 3 now what about these two lines do they have the same slope 1/5 1/5 yes so that means that they are parallel since they have the same slope so in this case we just have to change this form into the general form of a line which is y equals MX plus B now for the last one we have x equals 2 and sorry y equals 2 and x equals 3 for this one you`re not really going to use a y equals MX plus B method you just need to realize that if you ever have a line where the equation is just y equals a constant or x equals a constant that means it`s going to be either parallel or parallel to the y-axis or parallel to the x-axis so what do I mean let`s check y equals 2 y equals 2 well that would be this point here that`s where y equals 2 so y equals 2 there but also y equals 2 here and here and here so and here and here in here and here so the line y equals 2 is this horizontal line so this is the line y equals 2 now so that line is horizontal purely horizontal and it`s this is what I meant it`s parallel to the x-axis now let`s look at the line x equals three well x equals three is here x equals three but also x equals three is here and here and here and here and here and here and here so the line x equals three is a purely vertical line so this is x equals three and that line is parallel to the y-axis so this y equals two is parallel to the x-axis x equals three is parallel to the y-axis now what about the two lines itself they`re perpendicular one is purely vertical though there`s purely horizontal of course they are perpendicular so for these types of problems where they just equal constant they`re always perpendicular y equals 10 x equals fifteen thousand perpendicular and so on so these two lines alright that`s perpendicular and that is the answer hey great job guys thanks for watching the videos please be sure to check out our books on Amazon to specialize high school test prep book this book comes with five full-length exams with answer explanations along with a manual to go ahead and better prepare you for the exam you will find the link on the video or in the description above have fun studying