Finding The Slope Given 2 Points Tons of Examples

in this video we're going to focus on calculating the slope of line between two points so let's say if we have the point 2A 5 and the point 6 comma 13 what is the slope between these two points let's call this X1 and X2 and this is going to be y1 and Y2 the first number is the variable the second one is the variable now we can use this formula to calculate the slope it's Y2 - y1 / X2 - X1 so basically it's the change in divided by the change in now we Define Y2 to be 13 and y1 is 5 X2 is 6 X1 is 2 13 - 5 that's equal to 8 and 6 - 2 that's equal to 4 now we got to reduce it 8 / 4 is equal to 2 so this is the slope between these two points and that's how you could find it now let's work on some more examples try this one let's say if we have the point 3 -4 and also -5 pos2 so go ahead and take minute and calculate the slope between these two points so what like to do is identify the and variables and now let's use the formula so the slope is going to be Y2 - y1 / X2 - X1 so the second value is 2 the first value is -4 and then / X2 which is5 minus X1 which is posi3 2 --4 is basically the same as 2 + 45 minus pos3 is the same as5 - 3 2 + 4 is 65 - 3 is8 now we need to reduce the fraction both numbers are even so we can divide 6 and 8 by two half of 6 is three half of8 is4 so the slope is -3 divid 4 try these two examples 3A 6 and 5 comma 6 and also 41 and also 4 comma 8 go ahead and calculate the slope between each pair of points so let's start with the first one let's call this X1 and y1 and this is going to be X2 and Y2 so we know the slope is Y2 which is 6 minus y1 which is also 6 and X2 - X1 so that's going to be 5 - 3 6 - 6 is 0 and 5 - 3 is 2 0 ided by anything is equal to zero so the slope is zero whenever the slope is zero that means that you have basically horizontal line the horizontal line could be = 2 = 1 it doesn't really matter but it's horizontal line the slope for any horizontal line is always zero now what about the second example let's calculate the slope so is Y2 which is8 minus y1 which is ne1 / X2 - X1 which are both equal to four8 -1 is the same as8 + 1 whenever you have two negative signs next to each other you can change it into positive sign 4 - 4 is 08 + 1 is7 whenever you have zero in the denominator of fraction it's undefined we really don't have value for -7 / 0 so the slope all we can do is say it's undefined what does that mean in terms of graph basically the graph is vertical line vertical line has an undefined slope so if you were to plot these two points let's say the point 41 and also 48 41 would be somewhere over here and if we plot 48 it would be in this region so if you connect these two points of straight line you're going to get vertical line so the slope of any vertical line is always undefined sometimes you may have point that contains fraction in situation like this how would you find the slope between these two points go ahead and try so let's say this is X1 and y1 and this is X2 and Y2 so is the change in ID by the change in it's basically the rise / the run so in this problem we can see that Y2 is 1 5 y1 is 1 2 X2 is -1 over3 and X1 is 3 so how do we calculate the slope now that we have this complex fraction situation what you want to do in order to simplify complex fraction at least the best method think is to clear away all fractions so if we look at the smaller fractions we have denominator of 5 2 and 3 what is the least common multiple of 2 3 and 5 if you're not sure you can use any multiple it doesn't really have to be the least common multiple the quickest way to find multiple that would clear away the fractions is to multiply 2 3 and five you can write out list of numbers until you find common multiple but this is simple and fast technique 2 * 3 is 6 6 * 5 is 30 so what I'm going to do is I'm going to multiply the top of the fraction and the bottom of the fraction by 30 this will allow me to clear away all small fractions within the larger fraction so let's distribute what is 30 * 1 over 5 30 / 5 is six so 1/ of 30 is 6 half of 30 or or 30 / 2 that's 15 now we need to multiply the bottom part by 30 whatever you do to the top you must also do to the bottom 30 * 1/3 or 30 ID 3 is 10 but since we have 1/3 this is going to be -10 next 30 * -3 that's 90 and now let's simplify 6 - 15 is -9 and -10 minus 90 is 00 whenever you have negative number divided by negative number it's going to equal positive number so this is 9 / 100 so that is the slope of the line between these two points now for the sake of practice let's try another similar example so let's say if we have point 1 over 3 comma -1 over4 and another Point -2 over5 pos2 so go ahead and pause the video calculate the slope between these two points so first let's identify the points so is going to be 2 which is 12 minus y1 which is -4 / X2 which is -2 over5 minus X1 which is 1 over 3 now 12 -4 mean 12 - 1/4 that's 1 over2 + 1 over4 so now let's go ahead and simplify the complex fraction so what is multiple of 24 four five and three that we can use to clear away everything well we need five we need three and we need four we really don't need to multiply two and four because four is sufficient to get rid of two and four four is multiple of two so we don't really need to use eight 5 * 3 is 15 15 * 4 is 60 if you multiply 60 by 2 and use 120 that should be fine as well the only difference is that in the end you're going to have to simplify the fraction but you can still get the same answer but I'm going to use 60 60 is the least common multiple between 2 4 5 and 3 if you use the least common multiple you don't really have to simplify the fraction at the end if you use another multiple that's higher than the least common multiple then yes you're going to have to simplify now let's multiply the top and the Bottom by 60 so let's distribute 60 60 * half or 60 ID 2 that's going to be 30 and 60 ID 4 is 15 now what is 2 fths of 60 or -2 fths of 60 you can multiply and then divide or divide first like to divide first 60 / 5 is 12 and 12 * -2 is -4 so -25 of 60 is -4 1/3 of 60 or 60 ID 3 that's going to be 20 now 30 + 15 is 45 and -2 minus mean -4 minus 20 that's -44 so we can't really reduce this fraction so that's the answer -45 ID 44 and so this is it consider this problem let's say if the first point is 3 comma and the second point is 2 comma 5 and you're given the slope which is let's say 3 over 4 what is the value of how can we find the value of sometimes you may see question like this when doing your homework and to figure out the answerers not that bad first let's rewrite the formula is Y2 - y1 / X2 - X1 we have we know the value of X1 and y1 we don't know the value of take that back that's supposed to be X2 we don't know the value of y1 we have the value of Y2 so is 3 over 4 2 is 5 y1 is that's what we're looking for X2 is 2 X1 is 3 so basically we need to solve for the variable so let's cross multiply actually before we cross multiply we could simplify 2 - 3 2 - 3 is NE 1 so let's replace it with negative 1 now let's cross multiply 3 * -1 that's going to be -3 and that's equal to 4 * 5 - so we got to distribute the 4 4 * 5 is 20 4 * is4 now we got to solve let's subtract both sides by 20 -3 minus 20 is -23 and that's equal to4 so now we got to divide both sides by4 so is equal to 23 over 4 that's the answer here's another example - 12 comma 13 and comma 14 and let's say the slope between the two points is -5 / 3 What is the value of that's terrible question mark let's do that again so we're going to say this is X1 and y1 we'll look at it for X2 and we have Y2 so is the change in Yus the change in so we have is -5 3 2 is 1/4 y1 is 1/3 X2 is X1 that's going to be negative half so how can we find value what could we do in this problem well let's go ahead and cross multiply there's two things we could do we could cross multiply now or we could simplify the complex fraction and the order really doesn't matter but you know what let's simplify the complex fraction first so on the right side we have 14 - 1/3 and - half is the same as +2 so what should we multiply the top and bottom by to get rid of the three fractions well if we use four four that can clear away the four and two and we need three so let's use 12 let's multiply the top and the Bottom by 12 1/4 of 12 or 12 ID 4 that's 3 12 ID 3 is 4 and then * 12 is 12 half of 12 is 6 3 - 4 is -1 so we have -1 / 12x + 6 this is to that now let's cross multiply 3 * -1 is -3 and -5 * 12x + 6 we need to distribute -5 * 12x that's -60 X5 * 6 is -30 so let's add 30 to both sides -3 + 30 is 27 and now let's divide both sides by -60 so we have 27 over -60 which we can reduce it both numbers are divisible by three -27 / 3 is -9 60 ID 3 is 20 and that's as far as we can go so isal to 9/ 20 and so that's it now let's say if you're given linear equation what is the slope of this equation this equation is in the form = mx plus the slope intercept form the number in front of is the slope that's this number on the outside that's that's the intercept so the slope for this problem is two so try this one let's say if it's 5 - 23 what is the slope and what is the intercept now in this problem it's in the reverse order the slope is simply the number in front of so the slope is negative 2/3 you got to incorporate the negative sign as well the intercept is posi 5 is basically the constant without the now sometimes you might be given an equation that's in standard form in this example what is the slope of the line what we need to do is we need to get by itself we need to put it in slope intercept form so first let's move the 4X from the left to the right so let's subtract both sides by 4X so -2 is equal to -4x + 8 we can't combine these two because they're not like terms now let's divide every term by -2 to get by itself so is equal to -4 / -2 that's pos2 8 / -2 is4 so the slope is number in front of so it's equal to two