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Welcome to the presentation on systems of linear equations.

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

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So let's say I had two equations now.

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The first equation let me write it as 9x minus

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4y equals minus 78.

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And the second equation 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 is we're actually going to

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use both equations to solve for x and y.

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

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

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But now we have to equations.

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You can almost view them as two constraints.

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And we're going to solve for both variables.

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And you might be a little confused.

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How does that work?

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Is it just magic that two 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 rearranged each of these

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equations so that they 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 actual two equations because I

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

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

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

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

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

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

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

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second equation, 4x plus y 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 x's and y's that satisfy this

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equation, and on the green 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 x and y's that satisfy both

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equations, and you can guess where that is, that's

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right here right.

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

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

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

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this system of equations.

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

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

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if you can eliminate a variable 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 see, I want to eliminate, I

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

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an intuition for how we can do that later on.

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

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

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

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multiply this bottom equation 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 bottom equation by 4.

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

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Right?

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All I did is I multiplied both sides of the

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equation by 4, right?

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And you have to multiply every term because

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it's the distributive property on both sides.

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Whatever you do to one side you have to do to the other.

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Let me rewrite top equation again.

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

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track of things.

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9x minus 4y is equal to minus 78.

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

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add equations, you just add the left side and you

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add the right side.

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Well when you add, you 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 then we have minus 72 minus 78.

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So, let's see that's minus 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 sides by 25 or multiply both

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

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

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--x equals minus 6.

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There we solved the x-coordinate.

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Now to solve the y-coordinate 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, it seems a little bit,

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marginally simpler.

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So we just substitute the x back in there and we get

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4 time minus 6 plus y is equal to minus 18.

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Go up here.

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

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And then get y is 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 equations, you could even say,

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

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So they actually intersect someplace around here instead.

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

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But that's pretty cool, no?

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We actually solved for two 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 to do another problem.

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So let's say I had the points-- and I'm going to write them in

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

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and then the second equation is 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 fast as possible, I'd probably

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multiply this equation times 7 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 sometimes you might have to

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multiply both equations-- actually, not in this case.

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

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So let's multiply this bottom equation by 7.

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

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because I want this to cancel out with this.

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

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Let's write that first equation down again.

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Minus 7x minus 4y is equal to 9.

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Now we just add.

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

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OK, so that's 0.

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14 minus 4y plus 10y is equal to 30.

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y is equal to 3.

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Now we just substitute back 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 watch it again if you have.

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OK, so these two lines 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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Hope this one's harder.

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I think it will.

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

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We have minus 7x plus 4y 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 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 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 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 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 equation by 9.

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So it's minus 63x plus 36y is 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 two equations.

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

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264, let's see what's 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 solution manual or something.

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13 and 5, 70.

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

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

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375, and I know that seventy 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 both sides by 75.

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

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use this equation.

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

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We get minus 15 minus 9y equals 66.

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Minus 9y is equal to 81.

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And then we get y is equal to minus 9.

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So the answer is 5 comma minus 9.

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

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Have Fun.