❤︎² Gaussian Elimination.. How? (mathbff)

hey guys I'm Janelle and I'm going to show you how to use matrices in Gaussian elimination to solve a system of linear equations Gaussian elimination is like puzzle solving and every puzzle is going to be different I'm going to show you the rules and how to use them bad news is that there isn't a single set of steps that works every time but the good news is is that you're about to get better at this so let's do it before you can start solving systems of equations the first thing you have to do is rewrite the equations as a matrix so if you have some equations so it helps to write the equations so all the variables are in the same order just makes it easier to see what's happening I wrote these in the right order because I don't want to waste your time I'm just saying look out for that so let's make a matrix so each equation has three parts the coefficients the variables and the value that the equations equal we're basically going to copy each equation into a row of the matrix each column is part of the equation these columns are the coefficients for each variable and the last column will be the value of the equation equals basically we're just copying stuff here I'll start with the coefficients these are the coefficients from the variables if a variable doesn't have a coefficient the coefficient is actually just one because you have one of that variable that's why I wrote one here also if a variable is missing you put a 0 in its place because you don't have it so we're missing the Y variable here which is why I put the zero there now this last column these are the values the equations equal makes it easier if you put a line before the last column just to separate it out you don't have to it's not math or anything it's just help people remember what's going on now we have a matrix so what we even trying to do to it so how do you solve this well the goal is to make the matrix look like this I know this seems weird but if you make the matrix look like that you've basically solved the equations so where these dots are you'll have numbers but everything in the lower left corner needs to be 0 so Anna last row you should have all zeros except for two elements and as you go up each row has 1 left 0 in it so I'm sure you're wondering why because remember each column represents a coefficient for a variable from the equations so what we're really trying to do is reduce the equations which will make them super easy to solve later so now how do you make the matrix look like that so to get the matrix to look like what you want there are basically three things that you can do the first thing you can do is swap the rows around not the columns just the rows you can change your order around to try and get closer to what you want so here I swapped the last two rows second you can multiply a row by a constant so long as that constant isn't 0 you just multiply every element in the row by that constant so here I multiplied the second row by 2 and since fractions are constants too you can multiply a row by a fraction if you want to divide it finally you can add or subtract one row from another row you just add or subtract the elements in the same columns you can even add or subtract a multiple of a row this is really really useful and remember you're only changing one row when you do this now you know what you can do I'm going to show you the craziness and action so here are some equations I'm going to start by making the matrix to work on so remember the goal is to get everything in the lower left corner to zero if you can do that you can start substituting and solving so how do you know where to start well there isn't really any one right way to do this you try anything that you think will get you closer and if it doesn't work you can always go back and try something else if you see something where you can knock it out and get a zero where you need it that could be a good starting point like with the one here in the second row so I subtracted the first row from the second this got me a zero in the first column okay that was easy to spot now I'm going to do the same thing for the third row how don't forget you can add and subtract multiples of rows so by subtracting a multiple of a row I knocked out the first column of the third row so we're getting there so the last two rows have a zero in the first column this is really good because anything I do with just those two rows the result in the first column will stay zero so I added a multiple of a row again I had a two times the third row to the second row and now I row with two zeros in it so I can just swap that to the bottom and here we are we have the lower left corner zeroed so I did it in only a couple of steps because I didn't want to go crazy on you but it may take you more steps to get there I mean whatever as long as you get there you get there and I work this out so it stayed all nice with whole numbers but that's not realistic a lot of the time you're gonna end up with fractions but that's totally fine so it created a matrix from the original equations and now to solve I'm going to do the opposite I'm going to create three equations from the matrix so the columns left of the line are still the coefficients of the variables and so we get three new equations because of those zeros and the matrix some variables have been knocked out and now these are very solvable equations starting with the bottom equation will solve them working our way up so Z equals 1 and because there are only two variables in the second equation we can substitute the value of Z in to figure out the value for y so y equals 4 and finally we substitute y&z into the first equation to solve for X and that's it it was a long strange trip through the world of matrices but we solved all three equations I really hope this helped you understand how to do Gaussian elimination I know it's a lot to wrap your head around but just remember you don't have to do it alone and if you don't like math it's okay you don't have to but you can't like my video so if you did please like or subscribe

Channel: mathbff

Views: 59,374

Rating: 4.9247017 out of 5

Keywords: how to, how do you, matrix, matrices, solve, gaussian, elimination, gaussian elimination, equation, system of equations, scalar, element, entry, rows, columns, array, linear, algebra, algebra 2, linear algebra, jmt, math, mathematics, khan, how-to, help, solution, problem

Id: 1IHsX1lgpRI

Length: 9min 53sec (593 seconds)

Published: Mon Jul 06 2020