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Current time:0:00Total duration:8:07

Video transcript

we've already looked at several ways of writing linear equations you could write it in slope-intercept form what would be the form of Y is equal to M X plus B where m and B are constants M is the coefficient on this MX term right over here and M would represent the slope and then from B you're able to figure out the y-intercept the Y you're able to figure out the y-intercept from this literally the graph that represents the XY pairs that that satisfy this equation it would intersect the y axis at the point x equals 0 y is equal to b and its slope would be M we've already seen that multiple times we've also seen that you can also express things in point-slope form so let me make it clear this is slope-intercept slope-intercept and these are just different ways of writing the same equations you can algebraically manipulate from one to the other another way is point-slope point point-slope form and in point-slope form if you know that some if you know that there's an equation where the line that represents the solutions of that equation has a slope M slope is equal to M and if you know that x equals x equals a y equals B satisfies that equation that in point-slope form you can express the equation as Y minus B is equal to M over X or M times X minus a this is point slope form and we do videos on that but what I really want to get into in this video is another form and it's a form that you might have already seen and that is standard form standard standard form and standard form takes the shape of a X plus B y is equal to C where a B and C are integers and what I want to do in this video like we've done in the ones on point-slope and slope-intercept is get an appreciation for what a standard form good at and what is standard form less good at so let's give a tangible example here so let's say I have the linear equation it's in standard form 9x plus 16y is equal to 72 and we want it to graph this so the thing that standard form is really good for is figuring out not just the y-intercept y-intercept is pretty good if you're looking using slope intercept form but we can figure out the y-intercept pretty clearly from standard form and the x-intercept the x-intercept isn't so easy to figure out from these other forms right over here so how do we do that I want to figure out the x and y-intercepts let's just set up a little table here X comma Y and so the x-intercept is going to happen when Y is equal to 0 and the y intercept is going to happen when X is equal to 0 so when y is 0 what is X so when y is 0 16 times 0 is 0 that term disappears and you're left with 9 X is equal to 72 so if 9 times X is 72 72 divided by 9 is 8 so X would be equal to 8 so once again that was pretty easy to figure out this term goes away and you just have to say 9 times X is 72 X would be 8 when y is equal to 0 X is 8 so the point let's see Y is 0 X is 1 2 3 4 5 6 7 8 that's this point that right over here this point right over here is the x-intercept when we talk about x-intercept we're referring to the point where the line actually intersects the x axis now what about the y-intercept well we set x equals 0 this disappears and we're left with 16 y is equal to 72 and so we could solve we could solve that so we could say alright 16 y is equal to 72 and then divide both sides by 16 we get Y is equal to 72 over 16 and let's see what is that equal to that is equal to let's see they're both divisible by 8 so that's 9 / - or we could say it's 4.5 so when X is zero Y is four point five and so we could plot that point as well X is zero Y is one two three four point five and just with these two points two points are enough to graph a line we can now graph it so let's do that so let me whoops that I was using the tool that would draw a straight line let me see if I can so the line will look something like that there you have it I've just graphed I've just graphed this is the line that represents all the X&Y pairs that satisfy the equation 9x + 16 y is equal to 72 now I mentioned standard forms good at certain things and the good thing that standard form is maybe somewhat unique relative to the other forms we looked at is it's very easy to figure out the x intercept it was very easy to figure out the x intercept from standard form and it wasn't too hard to figure out the y-intercept either if we looked at slope intercept form the y intercept just kind of jumps out at you at point-slope form neither the X nor the y intercept kind of jump out at you the place where slope intercept or point-slope form are frankly better is that it's pretty easy to pick out pick out the slope here while in standard form you would have to do a little bit of work you could use these two points you could use the x and y intercept as two points and figure out the slope from there so you can literally say okay if I'm going from this point to this point my change in X my change in X to go from eight to zero is negative eight and to go from zero to four point five or the little Delta there unnecessarily let me so when you go from eight to zero your change in X is equal to negative eight and to go from zero to four point five your change in Y is going to be four point five so your slope once you've figured this out you could say okay this is going to be change in Y four point five over change in x over negative eight and since I at least I don't like a decimal up here let's multiply the numerator the denominator by 2 you get negative 9 over 16 now once again wait to do a little bit of work here we either use these two points it didn't just jump immediately out of this although you might see a little bit of a pattern of what's going on here but you still have to think about it as a negative is it positive you have to do a little bit of algebraic manipulation or what I typically do if I'm looking for the slope I actually might put this into into one of the other forms especially slope-intercept form but standard form on it by itself great for figuring out both the x and y-intercepts and it's frankly not that hard to convert it to slope-intercept form let's do that just to make it clear so if you start with 9x let me do that in yellow if we start with 9x plus 16 y is equal to 72 and we want to put it in slope-intercept form we can subtract 9x from both sides you get 16 Y is equal to negative 9x plus 72 and then divide both sides by 16 so divide everything by 16 and you'll be left with Y is equal to negative 9 16 X that's the slope you see it right there plus plus 72 over 16 we already figured out that's 9 halves or 4.5 so like it right well I'll just write that is 4.5 and this form over here much easier to figure out the slope and actually the y-intercept jumps out at you but the x-intercept isn't as obvious