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The Elimination Method

Added Feb 27, 2017 •

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here we go with another math is everywhere topic of the day we're timeout systems of equations and how to solve them here's one more method you can use to solve a system of equations when's it's called The Elimination method so we talked about graphing we talked about using the substitution method now we're going to do the elimination method we'll see why it's called the elimination method in just a second so  in the last video on the substitution method we talked about this problem there's two numbers we what the two numbers are but the sum of the two numbers is 40 and their difference is 4 so find the two numbers well if the sum of the two numbers is 40 that means they are 240 will call the tomb mystery numbers X and Y and whatever those two mystery numbers are their difference is 4 that means When You Subtract 1 from the other you get for so how do we find those two numbers will we could graph both these equations and see where the lines intersect we could use the subs Houston method but what we're going to use today is something called the elimination method and when we use the elimination method what we try to do as we try to take the two equations and line them up one right underneath the other and this works really well when both equations have terms lined up so that the X's are stacked up in the wise are stacked up and then the display number parts are stacked up there works really tell when we have it like that so watch what we can do when we do have are like terms all lined up we can basically just take this whole thing and we can  add this way X Plus Max's 2X and then add this way and then add this way but what happens when you do why plus negative Y is 1 y + -1 y actually cancels the wise and you see why it's called  the elimination method yes because we just eliminated the wise what do we have left well just to wax on the left side and 40 + 4 on the other side and watch out easy this is to solve you just divide sides by two and you got x equals 22 and we're halfway done with this problem now to solve this problem we know we have to have the why part so how do you find the white part will you take the 22 you plug it in four acts and we're going to do like a substitution thing here so 22  plus y equals 40 and now you just subtract 22 from both sides and you got why equals 18 and so now we have the solution to our system of equations 22, 18 or axes 22 why is 18 and the whole thing happened when we took our system of equations and we lined up the X's lined up the planes and lined up the other parts and then when we do we just added added added and figured out what the equation has that we have left so we're going to try that again so in case not example didn't exactly work for you will try another example what if you have X Plus y equals 17 + y equals 2x + 2 well when you try to line these up the elimination doesn't really work very good for a problem like this because did the like terms were line up with each other not really X and Y aren't like terms why into X aren't like terms so we wouldn't probably use the elimination method for a problem like this alright but for problem like this it might work because oh yeah look the EX's in the wise lineup so let's see what we get we're going to if we add and add an ad on this one  we're going to get 9 x + y + y is do y + 10 + 8 is 18 wait a second I thought this is called the elimination method something was supposed to get eliminated but we didn't liminate the wise do the access there still a why they're in there still an ex there so you know what we're going to do instead of adding  an adding and adding we're going to subtract so watch what happens when you subtract each time so we're going to take this whole equation and we're going to subtract now it will get when we line this up is  5 x -4 x gives us One X  and Y minus y gives us nothing there's our elimination that we were looking for I'm 10-8 gives us to everybody see how we did 5 x -4 x gives us one x y - y us nothing and 10-8 gives us to so now that we have that we actually solve for x because if one equals 2 that means x equals two and we know half of our solution is x equals to write what's the other half of our Solution that's we're going to plug the two in 4X right there  so we're going to do 5 x 2 instead of 5 x x plus y equals 10 and we know what 5 * 2 is its 10:10 plus y equals 1000 why must be nothing why must be Z what is 10 plus y equals 10 and now we've solved our whole system of equations and we have a solution okay so that was the elimination method and sometimes we subtract and subtract and subtract and sometimes we add add add each time it just depends what the two equations look like alright let's try another example if you have x + 2 y equals 10 + 3x + 4y equals 8  what's going to happen here is if you add that's not going to eliminate anything you're going to get 4X and if you adhere that's not going to eliminate anything you're going to get six y but if you subtract here  that would be like one x - 3x which is -2X and if you subtract here that would be like to y - 4 Y which would be -2 y. Didn't eliminate anything so does the elimination method always work not always perfectly easily but guess what there's a trick we can do on this we can take this whole first equation and since nothing eliminates yet we can make something eliminate so what really want is instead of an Axe and a 3X we want a 3X in a 3X or instead of a 2y and a for why we want to for why and a so what we're going to do is we're either going to try to make these access match or try to make these wise match so multiply this whole top equation by 3 and watch what I get 3x + 6y now because I distributed three here and I distributed the three there and then I'm going to have to distribute the three-year equals 30 and now what I can do is I can one take this equation and I can subtract it so I'm going to do 3x - 3x right - 4y  and then minus 8 and what I'm going to get here is 3 x -3 x cancels out there's my elimination that I wanted and then 6y - 4y is 2y 30 - 8 is going to give me 22 and now I can just finish solving this by dividing both sides by two those canceled and I get why was 11 now am I done no I only have the Y part I need to get the ex parte here and I can do that of course by just plugging something and I can plug this in right there because I know why is 11 so now I know that X + 2 * 11 equals 10  and I can just solve this equation this is 22 and I'm going to subtract 22 from both sides and then this will scratch out and I'll get x equals 10 - 22 make it a 12 and so now I know my solution to the whole system of equations is x equals -12 y equals 11 text so sometimes in order to do the elimination method you've got to take an entire equation and multiply it by three or X some other number okay I'm going to do a really fast new version of the same problem okay and I'm not not going to try to confuse you here so if you're getting just ignore what I'm about to say you could also instead of multiplying this by 3 you could also have multiplied this whole thing by two because watch what happens when you do that you go to x  and you got 2 x 2 Y is for y and then you do two times one is 20 and that might not seem like it helps until we see that we can subtract this whole equation and so we're going to go - 3x Arena - 4 y now we see what's going on tonight and we going to do -8 and so what we're going to get here is 2 x -3 Max is -1x + 4y - 4 white that's our elimination that's what we were looking for so we did a lemon something and now we get -1x equals 12 and I'm just going to divided by -1 so that those scratch divided by -1 and uh tax equals -12 and of course we'd have to continue in Sulphur why but what I'm trying to show you here is we could have made a match with these two to make an elimination or we could have made a match with those two to make an elimination either way gives us the elimination we're looking for in the grand elimination method so that is what we call the elimination method here's a practice problem for you to use and this one's on the simplest on the simpler and I think what you can do is  either add add add and get a new equation I'm here or you could subtract subtract subtract and get a new equation see which one gives you an elimination and helps you saw 4 X or Y if you're confused just ask me some questions tomorrow and class and we'll work on it together thanks for your time thanks for your patience thanks for all of your hard work in making sense of these problems and persevering and solving them  here's another practice problem if you want to stata try go go ahead and try go for it if you think that looks too scary don't worry about it we'll talk about in class tomorrow alright that's all I got for you play this video so long have a great day

Uploaded by Jeff Marbut.

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The Elimination Method

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