Gaussian Elimination with Back Substitution

in this video you're going to learn how to do Gaussian elimination with back substitution so this is just another technique for solving systems of equations instead of doing the graphing method or the elimination method of the substitution method now we're using basically like a matrix method and again it's called Gaussian elimination with back substitution where to go through two examples the first example you can see we've got three equations three variables our first step is to write it as an Augmented matrix so what is an Augmented matrix well essentially what an Augmented matrix is is you're just using the coefficients okay the numbers in front of the variables and you're also adding on the solutions here in this for this right column so what all I'm doing is I'm just writing down those coefficients writing down the answers okay and you can see I'm separating with little line or dashed line okay like so and the way that Gaussian elimination works is that you can interchange any two rows okay you can multiply any row by a constant as long as it's not zero and you can add any two rows together so let's see if I can show you how this works our first step is we're going to try to get 0 in this lower left-hand corner and the way I'm going to do that is going to multiply the first row by negative 1 and I'm going to add it to the third row I'm going to put the answer in the third row now I want to warn you right from the very beginning it's good to make a little note about what you're doing in case you make a mistake and you want to go back and track you know the steps that you did it's very easy to do there's a lot of just like little arithmetic errors that can happen but let me see if I can show you so negative 1 okay plus 1 gives us 0 negative 2 plus 4 is 2 positive 3 plus negative 2 is 1 and negative 1 plus 9 is 8 okay the other two rows are staying the same so I'm just going to copy those down as they were okay and let's see so now the next step is we want to get 0 in this position right here and the way I'm going to do that is I'm going to multiply negative 2 times Row one I'm going to add that to Row 2 and I'm going to put the answer in Row two ok so let's go ahead and do that now we're going to multiply negative 2 times 1 which is negative 2 plus 2 is 0 negative 4 plus negative 1 is negative 5 6 plus 1 is 7 and negative 2 plus 1 is negative 1 so you really have to be careful with arithmetic because if you make a little mistake it's going to throw the whole system off okay so now we've got zero in these two spots here our next step is to get zero right here okay and the way that I'm going to do that is I'm going to multiply the row 2 by 2 so 2 times Row 2 and then I'm going to add it to 5 times Row 3 I'm going to put the answer in Row 3 ok so they're with me so far so let's go ahead and do that so we're going to say 2 times Row 2 okay so that's going to be 0 and then this is going to be let's see negative 10 plus positive 10 that's going to give us 0 okay good and then this is going to give us 14 plus 5 is 19 right and then over here we've got negative 2 plus 40 which is 38 ok now the other roles they don't change they're staying the same so I'm just going to copy those down it's a lot of a lot of writing but the nice thing is you don't have to write the variables over and over again so now you can see we've gotten zeros here in this lower left-hand corner the next step is you want to get ones on the diagonal okay so you're with me so far you want to get ones on this diagonal so what I'm going to do is I'm going to multiply 119th times Row three I'm going to put that answer in Row three I'm also going to multiply negative 1/5 times Row two and I'm going to put the answer in Row two and then the first one is already starts with the 1 so we're good there so let me go ahead and do that next step here so that's going to give us let's see what we have here we've got 1 2 negative 3 1 ok and we've also got 0 1 and let's see negative 7/5 and we're multiplied by negative 1/5 so this is 1/5 and let's see what else do we have we've got 0 0 1 & 2 okay so now if you can see what we've got so far this we're going to rewrite these as equations now so this is actually the top equation is 1 X plus 2 y minus 3 Z equals 1 the second equation is 0 Oh X plus 1 y minus 7 5 z 7 v z equals 1/5 and the last one is 1 z is equal to 2 so now here's where the back substitution comes in you see this kind of stair step pattern we've got one variable one equation you know get the second equations got two variables the top equations got three variable so we're just going to back substitute so if I put two in for Z okay we already know what Z is we've got Y minus seven fifths times two equals 1/5 so that's going to be Y minus 14 fifths equals 1/5 and if we add the 14 fist to the other side we get 15 fists which equals 3 okay so now we know what Z and Y are we just have to solve for X let's go to the top equation we've got X plus 2 times 3 because that's what Y is minus 3 times 2 that's what Z is equals 1 so we've got X plus 6 minus 6 is equal to 1 the 6 and the negative 6 cancel so you can see x equals 1 and if you want to write your final answer you would write it as a triple X is 1 Y is 3 Z is 2 that's your answer and then you can check your answer if you want by taking you know these values x y&z putting them into all three of the equations here making sure that you know they give us the appropriate solutions on the right so again this is you know Gaussian elimination with back substitution we're going to go through another example to show you more how it works but before I do that if you're preparing for the AC T or the SAT you know check out my huge SAT math review video course on my huge AC t-- math review video course they go through a lot of different concepts we have a teaching segment there's opportunities to practice and do some problems and it just gives you a good overall review and an insight into some of the key things that they want you to know when you're taking that test so definitely check that out I have a link below in the description and also you can check out the links through my about page on my youtube channel Myers math tutoring but let's get into the next example let me erase this board and I'll be right back at you ok example number two we you can see we've got a three by three okay three variables three equations and the first thing we want to do is worried our Augmented matrix so let's go ahead and do that so we're just using the coefficients the numbers in front of the variable right and let's see 1 3 1:18 we've got some larger numbers here so we might have to take out the calculator to do some of these calculations okay to make it go little bit quicker but remember our first step here is to get 0 in this lower left-hand corner and the way we do that is we want to combine either the first with the third or the second with the third I'm just going to take negative 2 times Row 2 plus Row 3 I'm going to put the answer in Row 3 ok so if we do that I'm just going to copy down I usually like to kind of write down the rows that haven't changed just to kind of get that out of the way and so now this is the row that's going to be changing Row 3 so negative 2 times 1 is negative 2 plus 2 is 0 negative 6 plus negative 1 is negative 7 negative 2 plus negative 2 is negative 4 and negative 36 plus negative 3 is negative 39 okay so you can see the numbers are getting large here okay the next step we want to get 0 right here where that one is and so what I'm going to do is I'm going to combine the first equation with the second equation so that's going to be negative let's see negative 4 times Row 2 plus Row 1 answer goes in Row 2 alright so let's go ahead and do that so we've got let's see a copy down this bottom row because that's not changing yeah and again be careful that you don't make a little simple mistakes because that's easy to do with this method ok and so let's see so we've got negative 4 times this role plus this road so that's going to give us 0 negative 12 plus 1 is negative 11 negative 4 plus negative 1 is negative 5 negative 4 is going to give us negative 72 plus 2 12 is negative 60 okay so now we've got zero in this spot this spot now we want to get 0 over here where the negative 7 is and so the way I'm going to do that is I'm going to combine the second row with the third row now you might be thinking well why aren't you combining the first with the third well if I combine the first of the third what's going to happen is you're going to end up getting a constant besides zero in this lower left-hand corner which is going to kind of reverse out what we you know did earlier when we try to get 0 here so that's why I'm just combining the second in the third because you can see when you add 0 and 0 that's just going to still be 0 so it's not going to get rid of this 0 in this place here so let's go ahead and figure this one out we're going to have to let's see multiply 7 times Row 2 right plus a negative 11 times Row 3 and we're going to put the answer in Row 3 okay so let's go ahead and do that so we've got 4 we've got one we've got negative 1 we've got 12 right and let's see this one is going to stay the same so 0 negative 11 negative 5 negative 60 this one is going to be let's see 7 times Row 2 that's negative 77 plus positive 77 that's 0 we've got negative 35 okay plus 44 which is how much that's 9 right and then we've got let's see here's where I'm going to break out the calculator we've got 7 times Row 2 so that's going to be negative 4 20 right plus a negative 11 times a negative 39 which gives us 9 okay so that worked out nice ok so there we go so now you can see we've gotten zeros in that lower left-hand corner the next step is to get ones on the diagonal okay so the way I'm going to do that I was going to multiply 1/9 times Row three I'm going to put the answer in Row three I'm going to multiply Row two times negative one 11th and I'm going to put the answer in of course Row two and then Row one we're going to multiply by 1/4 and I'm going to put the answer in Row one so I'm just multiplying through by the reciprocal of the leading coefficient for that row so if I do that let me switch over here now okay we've got on the bottom row we've got 0 0 1 and 1 okay and in the second row we've got 0 1 let's see 5 11 and let's see this is going to give us 60 over 11 and then we've got one 1/4 let's see negative 1/4 and 3 okay so our equations now are going to be let's see we've got the top row which is X plus 1/4 Y minus 1 for Z is equal to 3 we've got 1 y plus 5 11 Z is equal to 60 11s right and then the last one we can see that we have Z equals 1 so we've already got Z solve for so that's good now we're going to do our back substitution so if I put 1 in for Z that gives us y + 5 11 s is equal to 60 11 s subtract the 5 11 s and that gives us Y is equal to 55 11 s which is equal to 5 ok so we know what Y is now now we have to solve for X let's put those other values in we've got X plus 1/4 times y which is 5 let's see plus a negative 1/4 Z which is 1 is equal to 3 so this gives us five fours - 4/4 which is negative 4/4 which is I said that wrong 5/4 minus 1/4 is 4 force which is 1 so that's X plus 1 is equal to 3 if we subtract 1 we get X is equal to 2 and if we write our final answer as a triple we've got X is 2 y is 5 and Z is 1 and that's our answer and again if you want to check your you know to make sure you did it right you can take these values put them in for XY and Z and all three equations and you know verify that you've done it correctly and that's exactly right so this is how to do the Gaussian elimination with back substitution I'm going to have another video where we do the Gauss Jordan elimination with the row reduced echelon form so if you're interested take a look at that video and go ahead and subscribe to the channel check out more math tutoring videos on my youtube channel Mario's math tutoring I look forward to helping you in the future videos and I'll talk to you soon

Channel: Mario's Math Tutoring

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Keywords: gaussian elimination, back substitution, gaussian elimination with back-substitution, what is gaussian elimination, gaussian elimination with back substitution, gaussian, solving systems of equatinos, row echelon form, free, math, maths, how to, mathematics, video, tutorial, mario's math tutoring, elimination, how to do gaussian elimination

Id: jOC4fMgl7TI

Length: 12min 44sec (764 seconds)

Published: Thu Jun 01 2017