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Welcome to the presentation on[br]systems of linear equations.

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So let's get started and[br]see what it's all about.

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So let's say I had[br]two equations now.

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The first equation let[br]me write it as 9x minus

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4y equals minus 78.

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And the second equation[br]I will write as 4x plus

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y is equal to mine 18.

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Now what we're going to do now[br]is we're actually going to

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use both equations to[br]solve for x and y.

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We already know that if you[br]have one equation, it has one

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variable, it is very easy to[br]solve for that one variable.

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But now we have to equations.

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You can almost view them[br]as two constraints.

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And we're going to solve[br]for both variables.

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And you might be a[br]little confused.

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How does that work?

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Is it just magic that two[br]equations can solves

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for two variables?

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Well it's not.

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Because you can actually[br]rearranged each of these

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equations so that they[br]look kind of in normal y

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equals mx plus b format.

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And I'm not going to draw these[br]actual two equations because I

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don't know what they look like,[br]but if this was a coordinate

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axis-- and I don't know what[br]that first line actually does

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look like, we could do another[br]model where we figured it out

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--but lets just say for sake of[br]argument, that first line all

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the x's and y's that satisfy 9x[br]minus 4y equals negative

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78, let's say it looks[br]something like that.

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And let's say all of the x's[br]and y's that satisfy that

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second equation, 4x plus y[br]equals negative 18, let's say

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that looks something like this.

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Right?

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So, on the line is all of the[br]x's and y's that satisfy this

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equation, and on the green[br]line are all the x's and y's

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that satisfy this equation.

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But there's only one pair of[br]x and y's that satisfy both

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equations, and you can guess[br]where that is, that's

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right here right.

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Whatever that point is-- I'll[br]do it in pink for emphasis.

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Whatever this point is,[br]notice it's on both lines.

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So whatever x and y that is[br]would be the solution to

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this system of equations.

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So let's actually figure[br]out how to do that.

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So what we want to do is[br]eliminate a variable, because

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if you can eliminate a variable[br]then we can just solve for

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the one that's left over.

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And the way to do that-- let's[br]see, I want to eliminate, I

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feel like eliminating this y,[br]and I think you'll get

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an intuition for how we[br]can do that later on.

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And the way I'm going to do[br]that is I'm going to make

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it so that when I had this[br]to this, they cancel out.

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Well, they don't cancel out[br]right now, so I have to

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multiply this bottom equation[br]by 4, and I think it'll be

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obvious why I'm doing it.

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So let's multiply this[br]bottom equation by 4.

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And I get 16x plus 4y is equal[br]to 40 plus 32 minus 72.

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Right?

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All I did is I multiplied[br]both sides of the

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equation by 4, right?

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And you have to multiply[br]every term because

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it's the distributive[br]property on both sides.

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Whatever you do to one side[br]you have to do to the other.

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Let me rewrite top[br]equation again.

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And I'll write in the same[br]color so we can keep

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track of things.

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9x minus 4y is[br]equal to minus 78.

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OK, well now, if we were to add[br]these two equations, when you

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add equations, you just add[br]the left side and you

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add the right side.

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Well when you add, you[br]have 16x plus 9x.

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Well that equals 25x.

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Right?

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16 plus 9.

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4y minus 4, that just equals 0.

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So that's plus 0 equals, and[br]then we have minus 72 minus 78.

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So, let's see that's minus[br]150, minus 150, right?

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Just adding them all together.

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So we have 25x equals 150.

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Well, we could just divide both[br]sides by 25 or multiply both

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sides by 1/25, it's[br]the same thing.

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And you get x equals--[br]that's a negative 150

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--x equals minus 6.

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There we solved[br]the x-coordinate.

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Now to solve the y-coordinate[br]we can just use either one of

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these equations up at top.

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So let's use this one,[br]it seems a little bit,

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marginally simpler.

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So we just substitute the x[br]back in there and we get

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4 time minus 6 plus y[br]is equal to minus 18.

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Go up here.

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4 times minus 6 we get minus 24[br]plus y is equal to minus 18.

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And then get y is[br]equal to 24 minus 18.

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So y is equal to 6.

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So these two lines or these two[br]equations, you could even say,

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intersect at the point x is[br]m inus six and y is plus 6.

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So they actually intersect[br]someplace around here instead.

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I drew these, the line probably[br]look something more like that.

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But that's pretty cool, no?

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We actually solved for two[br]variables using two equations.

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Let's see how much time I have.

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I think we have enough time[br]to do another problem.

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[br]105[br]00:05:20,2 --> 00:05:23,02[br]So let's say I had the points--[br]and I'm going to write them in

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two different colors again[br]--minus 7x minus 4y equals 9,

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and then the second equation is[br]going to be x plus

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2y is equal to 3.

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Now if I were doing this as[br]fast as possible, I'd probably

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multiply this equation times 7[br]and it would automatically

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cancel out.

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But that's easy way.

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I'm going to show you that[br]sometimes you might have to

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multiply both equations--[br]actually, not in this case.

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Actually let's just do it[br]the fast way real fast.

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So let's multiply this[br]bottom equation by 7.

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And the whole reason why I want[br]to the, multiply it with 7,

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because I want this to[br]cancel out with this.

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If you multiply it by 7 you get[br]7x plus 14y is equal to 21.

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Let's write that first[br]equation down again.

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Minus 7x minus 4y[br]is equal to 9.

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Now we just add.

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This is a positive 7x, it just[br]always looks like a negative.

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OK, so that's 0.

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14 minus 4y plus 10y[br]is equal to 30.

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y is equal to 3.

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Now we just substitute back[br]into either equation,

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lets do that one.

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x plus 2 times y, 2 times 3.

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x plus 6 equals 3.

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We get x equals negative 3.

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That one was super easy.

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The intercept.

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Hope I didn't do it to fast.

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Well, you can pause it and[br]watch it again if you have.

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OK, so these two lines[br]intersect at the point

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negative 3 comma 3.

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Let's do one more.

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[br]140[br]00:07:07,456 --> 00:07:10,71[br]Hope this one's harder.

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I think it will.

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OK, negative 3x minus[br]9y is equal to 66.

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We have minus 7x plus 4y[br]is equal to minus 71.

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So here it's not obvious.

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What we have to do is, let's[br]say we want to cancel

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out the y's first.

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What we do is we try to make[br]both of them equal to the least

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common multiple of 9 and 4.

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So, if we multiply the top[br]equation by 4 we get--

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I'll do it right here.

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Let's multiply it by 4.

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Times 4.

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We'll get minus 12x minus[br]36y is equal to 4 times

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240 plus 24 is 264.

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Right, I hope that's right.

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We multiply the second[br]equation by 9.

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So it's minus 63x plus 36y is[br]equal to, let's see, 639.

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Big numbers.

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639.

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OK, now we add the[br]two equations.

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Minus 12 minus 63 thats minus[br]75x-- these cancel out --equals

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264, let's see what's[br]639 minus 264.

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See I do this in real time.

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I don't use some kind of[br]solution manual or something.

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13 and 5, 70.

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I don't know if I'm[br]right, but we'll see.

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Since it's actually the[br]negative 639, this is minus

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375, and I know that seventy[br]five goes into 300 4

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times, so x is equal to 5.

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75 times 5 is 375.

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We just divided[br]both sides by 75.

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So if x is 5 we just substitute[br]it back into-- let's

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use this equation.

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So we get minus 3 times 5[br]minus 9y is equal to 66.

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We get minus 15[br]minus 9y equals 66.

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Minus 9y is equal to 81.

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And then we get y is[br]equal to minus 9.

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So the answer is[br]5 comma minus 9.

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I think you're ready to do some[br]systems of equations now.

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Have Fun.