College Algebra_H: Ch.2.3 Equation in Slope-Intercept format, Graph by slope- intercept, Int

foreign okay question number one find the slope of the passing uh slope of the line passing through the points okay through the points 2 and 4 and three and six okay through the points two and four and three and six uh and we need to find the slope and then when we need to indicate whether the line that goes through this point rises false or horizontal or is vertical so before we start solving this i would like to give you some general understanding so slope we usually use the letter m for slope and slope is equal y2 minus y1 x2 minus x1 so in this case we will have here x1 y1 because when we have coordinate first number is always nailed for x1 second number is nailed for y1 and here x2 and y2 now can could i switch in 2 and 4 take so x2 y2 and 3 and 6 take as x1 y1 absolutely there will be no difference whatsoever so you still would end up with the same answer okay so m is equal y2 minus y1 x2 minus x1 this is definition of the slope okay next if we get slope positive greater than zero then the graph or uh the uh yeah of the line is rising if we get slope negative then the line is falling when m is zero then we have flat horizontal line more down and if we have m is equal infinity then the line is going vertically okay and the other thing we i need to write down the format of slope intercept slope intercept is y is equal mx plus b where m is your slope and b is your y-intercept okay so we can start working on this problem so we accepted that um we at the have 2 and 4 is x 1 y 1 3 and 6 is x 2 y 2 and our goal is to find slope so m is equal based on formula we have to do this formula we have to do y 2 minus y 1 y 2 is 6 oops 6 minus 4 and three minus two okay this will gonna be equal to over one so which is going to be two and our slope is 2 right since m is greater than 0 which is positive 2 then our graph is our graph is rising right okay question number two the same task for the given points which is we need to find the slope and identify if the line i that goes through these points is uh uh point rises or false or horizontal or vertical so we have points negative one and four and we have point six and two okay you make space so i can write down so in this case i'm going to use this one as a x2 y2 and this one is a x1 y1 so to show you that it doesn't matter which one will be set of x2 y2 which one would be set of x1 y1 so let's find out our m the m still follows same way [Music] i can take y 2 4 minus y 1 which is 2. i can take x 2 negative 1 minus x 1 minus 6. so we get 2 over negative 7 so our slope so the slope is less than 0 which means the graph is gonna fall right by the way something came to my mind and i uh i wanna bring to your attention this definition of slope so instead of y2 minus y1 x2 minus x1 it also can be y1 minus y2 x1 minus x2 doesn't matter what is important so if the first term let me see if the first term here you take you start with y2 then the bottom also has to be y2 okay second term if you have white 1 then x also has to be y1 the index you have to keep the same see here if i started y1 in denominator also i started with x1 here i subtracted y2 in denominator also x2 that's what is important otherwise you will do a y2 minus y1 or y1 minus y2 it's not important you will get the same answer just keep the index if you started with y2 in denominator you start with x2 if you started with y1 in numerator then denominator you start with x1 i hope i made myself clear so question number three now we have we need to write the point-slope form of the line so in this case we have m is equal three so we're changing given data slope is given so we need to find practically the equation that represent the line line given slope and point negative five and five okay so what we're going to do in this case i'm using again m is equal y minus y1 x minus x1 so if you are wondering why i didn't put y2 minus y1 x2 minus x1 for very simple reason that i have only one coordinate here x1 and y1 so that's why i kept y and x general form and here x1 and y1 okay so from here i can insist since we have equation in the denominator we have with x minus x one so to get rid of the denominator then i have to do here also x minus x one to bring to common denominator right in the same change to numerator okay i drop this and as a result i'm going to get m times x minus x1 equal y minus y1 and this is important formula i'm going to work with every single time i am given slope and one point okay okay so now what i'm going to do i'm going to plug m x1 and y1 our m is free x minus x1 is negative 5. equal y minus my y one is five okay now um let us simplify yes we get three times x plus five is equal y minus five now this form this form of equation is a slope point form slope or point slope [Music] form point slope form this is too thick let me make it thinner okay point-slope form this one now we but we need to get our equation um in the format of slope intercept which is this one okay slope intercept so how we're going to get just keep simplifying and getting y so if i move uh 5 negative 5 on this side remember every time i move any term over this equal sign i'm going to flip the sign it was negative 5 will become positive 5. so 3 times x plus 5 plus 5 is equal y so let's uh simplify 3x plus 15 i am distributing yes plus 5 is equal y and 15 and 5 like terms 3x plus 20 is equal y or y is equal 3x plus 20. does it looks like our slope intercept like y is equal let me write down here mx plus b so my b is 20 and here m is three okay so this is my slope intercept form intercept okay question number four let me write down here same thing m is equal negative 6 and our point is negative 2 negative 7 negative two all right let's go over we need to um write the equation for the line in the form of point slope intercept well let's get going so i'm using the format m times x minus x one equal y minus y one m is negative six x minus x one is negative seven equal y minus negative two okay because this is my x one this is my y one okay so let's simplify we will get negative 6 times x plus 7 equal y plus 2 because we have negative negative so it gives you plus and this is point slope format and then we keep simplifying so i'm distributing negative six we get negative 6x minus 42 is equal y plus 2. moving these two on this side which is becoming negative so negative 6x minus 42 minus 2 is equal y combining negative 6x minus 44 is equal y so from here rewriting y is equal negative 6x minus 44 and this is our slope intercept okay moving to question number five so what do we have in this case we have two points negative three negative six and we're back to two points again two and nine so using this uh two points we need to write an equation for the line in point slope form and slope intercept form okay so uh now given two points i have to write down the same two these forms that we have here this form and this form right so the only difference instead of while here we had a slope and point here we have two points which not a big deal because given two points we know how to find slope correct okay slope is equal oops m is equal now this time i'm writing y1 minus y2 and x1 minus x2 okay this is x1 it's too thick i don't like it x1 y1 x 2 y 2 okay so it's going to be equal y 1 is what negative 6 minus 9 and negative 3 minus two result is negative 15 negative five the answer is three so the slope is negative guess what the line is falling right because it is negative okay so slope is uh we got three now we back to this form where we have slope and point slope and point right the question is which point we have two of them we have here two points which one either one you can pick either one slope and either point so since working with positive numbers is more pleasant so i'm going to choosing to work with x2 and y2 okay so therefore i am using the format m times x minus x2 equal y minus y2 remember that form here okay so we have m is three x minus x two is two equal y minus y two is nine very good this is going to be in our point slope form okay now let's simplify further 3x minus 6 equal y minus 9 so if i move this negative 9 here will be positive 3x oops 3x minus 6 plus nine is equal y three x plus three like terms is equal y or rewrite y is equal three x plus three we are getting our slope intercept form okay question number six question number six another one like this so we have negative seven ten and we have six and ten given two points write down the equation of the line into form point slope form and slope intercept form okay let's get going so i'm going to uh look uh on this little at the notation on the example five and so the exactly same way because we are going to do same thing okay so m is equal i'm going to do 10 y 1 minus y 2 10 minus 10. and what do we have here negative seven minus six we are getting interesting information here zero over negative 13 which is equal zero bingo when we have m is zero remember let's go back here m is equal to zero we have horizontal line there is no slope yes so we have horizontal line okay now let's plug plug to the second uh format okay so we have m times x minus x1 let's take this time x1 y minus y1 and what do we get 0 times x minus negative seven and y minus ten why because this is my x one this is my y1 yes and as a result i'm going to get 0 times x plus 7 equal y minus 10 and this is our also point slope format let's simplify to get our slope intercept format so zero times whatever i don't even care what is in parenthesis it's going to be 0 correct so what do we have 0 is equal y minus 10 from here if i move 10 on this side i will get positive 10 right because was negative flipping over equal sign i get positive so 10 is equal y or y is equal 10 and this is going to be our equation why because for the format we have y is equal mx plus b if my m is 0 as we got here then mx is going to be 0 and we will get y is equal b so which means b 10 is our [Music] y y-intercept and this is our slope intercept format let me move it here also okay moving to question number seven so what do we have here give the slope and y-intercept of the line whose equation is given okay now we are going backwards before it was giving us slope and intercept we need to find the equation now we are given equation which is f of x is equal four over five x minus 7 we need to find slope and intercept okay so the slope is our m which is coefficient of our x and this is going to be our slope four over five so m is equal four over five okay and y-intercept this is our y-intercept it's going to be equal negative seven okay now we need to build the graph let's see they want us to build this graph how we're gonna build the graph uh based on given equation with the method of slope intercept okay we can do it it's not a big deal let's do it and show that we know how to do that okay okay we always start with the graph graphing by slope intercept format we always start with y-intercept which is negative seven the negative seven y-intercept meaning is a coordinate zero negative seven which is going to be down here this is our y axis this is x axis right so this is negative 1 here 1 2 3 4 5 6 7 negative 7 so this is our first point right here next and next i'm working with my slope slope is equal 4 over 5 which means the upper number which is number in denominator let me move it here so i can show you number in the enumerator it's running vertically either up or down based on positive or negative and the number in denominator always running horizontally okay either right or left based on uh sign since both of them are positive it's very easy to figure out so i'm going to go up by 4 but i'm going to go up from this y-intercept so if i go up by 5 by 4 1 2 3 4 i came here so guess what number is going to be negative 7 plus 4 we got to negative 3. now let's go here we have to go by positive five horizontally so let me break down one two three four five so from here i'm going to go by five and now we have our points first point is our intercept second point will show our slope and the graph is going to go through these two points let me change the color so you can see this is our graph okay taking too much space there we go question number eight we are going to do exactly same task given f of x going backwards now equation is given we need to find slope and intercept from the equation so negative 4 over 5 five x and plus eight okay first of all our m is a coefficient of x negative four over five and y-intercept is eight or rather saying coordinate zero and eight i would like to invite your attention on something not in all cases slope is going to be negative 4 over 5x only in the case of if equation is given in the slope intercept format that's why in previous problems we learned to bring equation to slope intercept format like this one very thick let me make it much better slope intercept format okay now if i would have let's say equation negative 3x plus 2y equal 5 then you do not dare to insist that negative 3 slope it's not you have to have so in this case what you have to do you have to find the y from here which is going to be 2y is equal i move negative 3x will be 3x plus 5 and divide everywhere by 2 and i would get y is equal 3 over 2x plus 5 over 2. see i got the format of y is equal mx plus b and only then i can say that 3 over 2 is uh our slope only in this format are we clear okay let me clean this all right so we continue working we have we have our slope and we have our intercept now let's graph it we are going to graph okay so now my y-intercept is eight which is coordinate zero eight one two three four five six seven eight here is interesting thing m is equal negative 4 over 5. so where this negative is going when we have negative in front of a numerator i mean in front of fraction i can read negative 4 over 5. i can read this of 4 over negative 5. so either way it's gonna work so i'm since my i have here eight if i take uh this four four positive then from eight i have to go even higher and i don't have space and i'm not interested to make a big graph so rather i'm going to take this one negative four over five so i'm going to go down by four okay so let me mark down one two three four five this is five so i'm going to go down by four so here was eight this eight and i do minus four i will get to four one two three four here i would get to this four or go down one two three four okay and then i have to go positive five horizontally and now i have my second point first point is y intercept second point is giving me slope and the result is going to be falling graph of falling line all right okay moving to question number nine so we are given y is equal three over seven x now if previously we had here uh f of x is equal see here also we have f of x it doesn't matter f of x and y it's the same thing because y depends on x so it's the same thing so don't allow this notation confuse you so what do we need to do give the slope and the intercept of the line with the given equation okay then the graph okay so we're going to do the same thing since we don't have any available terms so it's that's mean it's zero right so our slope is going to be three over seven and y-intercept it's going to be 0 because we don't have one now now let's uh build the graph how we're going to build the graph in this case very easy since our graph is zero i mean our y-intercept is zero we start from zero that's where y is equal zero and then i have m three over seven so i'm going up by three one two three and then horizontally by seven one two three four five six seven there we go and my graph is going to be interesting observation every time when your y-intercept is zero your graph is going to go through the origin through the zero okay because y-intercept is the one that shifts your graph up or down so we don't have y-intercept therefore the graph is going through [Music] center through zero okay question number ten x is equal four x is equal four let's uh graph it it's asking to graph so one two three four all right my x is equal 4 there is no y right absolutely y doesn't exist that's why my graph will never cross this y and it's going to run vertically which in what case it won't cross when it's running parallel correct and that's exactly what's going to happen this is my x is equal four okay next question number 11 i'm going to have let me move it here i can squeeze the other one also because it's very simple things cases question number eleven now we we need to graph y is equal seven guess what how it's gonna graph uh run it's gonna run horizontally right because now x is missing which means graph can never cross your x-axis and to do that you have to use your graph that will run parallel to your ex okay you wanna know what let me move it here down so i need seven so it's gonna be one two three four five six seven all right where is my graph this will be y is equal seven this is going to be my graph okay next question number 12. so rewrite uh the equation 5x plus y so we have equation 5x plus y minus 4 equals 0. so we need to write this equation in the slope intercept format remember slope intercept format was y is equal mx plus b which means i have to find y from here so let's see y is equal i'm going to move negative 4 on this side i'm going to move 5x on this side and it's going to flip the sign so 5x was positive it's becoming negative 5x negative 4 will become positive 4. so this is our slope-intercept format okay that's question a and then we have question b let's see what it is about um find the slope and intercept okay so from here it's easy right i m is equal how much you're right negative five and the intercept y-intercept is equal how much of course for available term all right and now we need to build the graph we can do that we already know how to do it right so y intercept is four one two three four this is first point we got it now it's interesting m is equal negative five so i'm going to represent as a fraction negative five over one yes okay uh and i'm going to use instead of negative five over one i'm going to go positive five over negative one so i'm going to use m why i'm doing this because i already left enough space here on the y axis to move up that's all i'm going to use five over negative one so i'm going to go up by five so one from this point right from intercept so if i had four if i go up by five will be nine correct one two three four five and then i'm going negative one which is coming this way negative one and here is my second point and my graph is this this is my graph okay another case like that question number 13. so equation we have 2x plus 7y minus 14 is equal 0. so every time when you need to graph equation like this in the form of slope intercept you always have to bring to the format y is equal mx plus b which we will do that 7y is equal negative 2x plus 14 right okay dividing by seven because we need y right and what do we get this is one this is two my y is going to be equal negative two over seven x plus two okay i broke to the slope intercept format let's graph it yeah what am i doing so my y-intercept is positive 2 1 2 one point is here now i'm using negative 2 over 7 so i'm going down this is two going down by two one and two which come to the zero and then going seven one two three four five six seven okay so it's gonna be right here right so the graph is gonna look like this okay and this one let me see if i can stretch it there we go okay question number 14. now here it's gonna use different method of graphing so till now we were graphing based on slope and intercept now there is another method it's a method of intercepts method of two intercepts let's do it question number 14 we have equation 7x minus 4y plus 28 is equal zero why wrote down so big i don't know okay better so how we're going to work with intercept it's going to be x-intercept y-intercept so to get um x-intercept i'm going to take i'm going to explore two cases when y is equal to zero how much will be x and when x is equal zero how much will be y those are method of two intercepts so when y is equal zero we get seven x plus 28 equals 0 okay and then from here 7x is equal negative 28 right i moved on this side and x is equal negative four so the first intercept is going to be let me move here is going to be x is negative 4 y is 0. this is my first intercept and the second one when x is equal 0 i have 7x will become 0 obviously from here it because 7 times 0 is going to be 0 and i will have negative 4y plus 28 equals 0 okay so from here negative 4y equal negative 28 and i need to divide by negative 4 here negative 4 here this is gone this will be 7 and my y is equal positive 7 right so the second intercept it's going to be 0 and 7. those are two intercepts that we were looking for so every time it will ask you to build a graph based on intercepts so you take x is equal to 0 find out how much will be y and then you take y is equal zero how much will be your x now uh let's uh build the graph let me move a little bit higher i can do that so okay so first point negative 4 0 so first one is x so one two three four negative four and y is zero so one point is right here second point we have zero seven so 7 is my y here so i'm going 1 2 3 4 5 6 7. this is my second point and i'm building the graph approximately going through the point this is my graph question number 15 another one using uh intercepts i pick two of each type so we can practice and you feel more comfortable with working with this method also we have 5x minus 3y minus 15 is equal zero okay we are using method of intercepts again it doesn't matter which one you take uh first x is equal to 0 or y is equal 0. so let's take this time x is equal to 0. when x equals zero five x is gone will be zero and i will have negative three y minus fifteen equals zero from here negative three y is equal positive fifteen and then we divide by negative 3 negative 3 and my y is going to be negative 5. so my first coordinate is 0 negative 5. now y is equal to zero so in that case here negative three three y was going to be zero yes and we will have five x minus 15 is equal zero 5x is equal 15 and x is going to be equal three so i'm getting three and zero okay let's build the graph well okay i did by hand already never mind so first point is zero negative five so negative five is my y one two three four five right here second point three and zero one two three and zero here we go and the graph is right here so this is negative five this is three okay question number sixteen we have two more problems and i will be done practicing again intercept so in this case we have 5x plus 5y and it's equal negative 25. so let's go x is equal 0 we will have left 5y is equal negative 25 and y is equal negative 5 from here so first coordinate will be 0 negative 5. i'm going a little bit faster because we solve already two problems and y is equal 0 so 5y is gone right from equation and we get 5x is equal negative 5 negative 25 i'm sorry and x is equal negative five which means coordinate will be negative five and zero okay so the graph gonna be like that so zero negative five and negative five zero let me go zero negative five 1 2 3 4 5 negative 5 this is one point and the other point is negative 5 0 1 2 3 4 5. negative five okay in the last question so 4x minus 2y equal negative 4 so x is equal 0 we have left negative 2y is equal negative 4 dividing negative two negative two and our y is equal positive two so zero positive two and second one y is equal to zero we have left four 4x is equal negative 4 and x is equal negative 1. so we have negative 1 and 0 building the last graph and we are done so 0 and 2 1 and 2 right here and negative 1 here is negative 1 so this is 2 this is negative 1 and my graph is this okay i'm done now at the end uh i would like to explain to you when which method you want to use so it all depends on the what type of equation you have what type of numbers let's say if i have negative 5x plus 7y is equal 12. look at that so it's not convenient to use method of intercepts because 12 divided by 5 is fraction 12 divided by 7 is fraction right so it's not convenient in this case it's uh better to use slope intercept method right but if you have let's say 6x minus 4y equal 24 look at that 6 4 24. you can divide 24 m by 6 and by 4 then it's easier to use method of intercepts so it all depends on numbers in the equation okay that's how you if it fraction if you have fraction then it's easier to work with slope intercept if numbers talk meaning you can divide like this one 24 4 and 6 then it's better use a method of intercept okay thank you for your patience and watching and i hope you are learning um give me sign that you like it uh if i would keep going this way put the like sign put the like there so i will know that you are learning you are enjoying all is good okay thank you i appreciate it it's following me so i'll see you in next video bye for now

Channel: Nara's Math Channel

Views: 323

Keywords:

Id: M3fKTBu1mOo

Length: 62min 15sec (3735 seconds)

Published: Thu Nov 19 2020