Old video on systems of equations systems of equations
Old video on systems of equations
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- Welcome to the presentation on systems of linear equations.
- So let's get started and see what it's all about.
- So let's say I had two equations now.
- The first equation let me write it as 9x minus
- 4y equals minus 78.
- And the second equation I will write as 4x plus
- y is equal to mine 18.
- Now what we're going to do now is we're actually going to
- use both equations to solve for x and y.
- We already know that if you have one equation, it has one
- variable, it is very easy to solve for that one variable.
- But now we have to equations.
- You can almost view them as two constraints.
- And we're going to solve for both variables.
- And you might be a little confused.
- How does that work?
- Is it just magic that two equations can solves
- for two variables?
- Well it's not.
- Because you can actually rearranged each of these
- equations so that they look kind of in normal y
- equals mx plus b format.
- And I'm not going to draw these actual two equations because I
- don't know what they look like, but if this was a coordinate
- axis-- and I don't know what that first line actually does
- look like, we could do another model where we figured it out
- --but lets just say for sake of argument, that first line all
- the x's and y's that satisfy 9x minus 4y equals negative
- 78, let's say it looks something like that.
- And let's say all of the x's and y's that satisfy that
- second equation, 4x plus y equals negative 18, let's say
- that looks something like this.
- So, on the line is all of the x's and y's that satisfy this
- equation, and on the green line are all the x's and y's
- that satisfy this equation.
- But there's only one pair of x and y's that satisfy both
- equations, and you can guess where that is, that's
- right here right.
- Whatever that point is-- I'll do it in pink for emphasis.
- Whatever this point is, notice it's on both lines.
- So whatever x and y that is would be the solution to
- this system of equations.
- So let's actually figure out how to do that.
- So what we want to do is eliminate a variable, because
- if you can eliminate a variable then we can just solve for
- the one that's left over.
- And the way to do that-- let's see, I want to eliminate, I
- feel like eliminating this y, and I think you'll get
- an intuition for how we can do that later on.
- And the way I'm going to do that is I'm going to make
- it so that when I had this to this, they cancel out.
- Well, they don't cancel out right now, so I have to
- multiply this bottom equation by 4, and I think it'll be
- obvious why I'm doing it.
- So let's multiply this bottom equation by 4.
- And I get 16x plus 4y is equal to 40 plus 32 minus 72.
- All I did is I multiplied both sides of the
- equation by 4, right?
- And you have to multiply every term because
- it's the distributive property on both sides.
- Whatever you do to one side you have to do to the other.
- Let me rewrite top equation again.
- And I'll write in the same color so we can keep
- track of things.
- 9x minus 4y is equal to minus 78.
- OK, well now, if we were to add these two equations, when you
- add equations, you just add the left side and you
- add the right side.
- Well when you add, you have 16x plus 9x.
- Well that equals 25x.
- 16 plus 9.
- 4y minus 4, that just equals 0.
- So that's plus 0 equals, and then we have minus 72 minus 78.
- So, let's see that's minus 150, minus 150, right?
- Just adding them all together.
- So we have 25x equals 150.
- Well, we could just divide both sides by 25 or multiply both
- sides by 1/25, it's the same thing.
- And you get x equals-- that's a negative 150
- --x equals minus 6.
- There we solved the x-coordinate.
- Now to solve the y-coordinate we can just use either one of
- these equations up at top.
- So let's use this one, it seems a little bit,
- marginally simpler.
- So we just substitute the x back in there and we get
- 4 time minus 6 plus y is equal to minus 18.
- Go up here.
- 4 times minus 6 we get minus 24 plus y is equal to minus 18.
- And then get y is equal to 24 minus 18.
- So y is equal to 6.
- So these two lines or these two equations, you could even say,
- intersect at the point x is m inus six and y is plus 6.
- So they actually intersect someplace around here instead.
- I drew these, the line probably look something more like that.
- But that's pretty cool, no?
- We actually solved for two variables using two equations.
- Let's see how much time I have.
- I think we have enough time to do another problem.
- 105 00:05:20,2 --> 00:05:23,02 So let's say I had the points-- and I'm going to write them in
- two different colors again --minus 7x minus 4y equals 9,
- and then the second equation is going to be x plus
- 2y is equal to 3.
- Now if I were doing this as fast as possible, I'd probably
- multiply this equation times 7 and it would automatically
- cancel out.
- But that's easy way.
- I'm going to show you that sometimes you might have to
- multiply both equations-- actually, not in this case.
- Actually let's just do it the fast way real fast.
- So let's multiply this bottom equation by 7.
- And the whole reason why I want to the, multiply it with 7,
- because I want this to cancel out with this.
- If you multiply it by 7 you get 7x plus 14y is equal to 21.
- Let's write that first equation down again.
- Minus 7x minus 4y is equal to 9.
- Now we just add.
- This is a positive 7x, it just always looks like a negative.
- OK, so that's 0.
- 14 minus 4y plus 10y is equal to 30.
- y is equal to 3.
- Now we just substitute back into either equation,
- lets do that one.
- x plus 2 times y, 2 times 3.
- x plus 6 equals 3.
- We get x equals negative 3.
- That one was super easy.
- The intercept.
- Hope I didn't do it to fast.
- Well, you can pause it and watch it again if you have.
- OK, so these two lines intersect at the point
- negative 3 comma 3.
- Let's do one more.
- 140 00:07:07,456 --> 00:07:10,71 Hope this one's harder.
- I think it will.
- OK, negative 3x minus 9y is equal to 66.
- We have minus 7x plus 4y is equal to minus 71.
- So here it's not obvious.
- What we have to do is, let's say we want to cancel
- out the y's first.
- What we do is we try to make both of them equal to the least
- common multiple of 9 and 4.
- So, if we multiply the top equation by 4 we get--
- I'll do it right here.
- Let's multiply it by 4.
- Times 4.
- We'll get minus 12x minus 36y is equal to 4 times
- 240 plus 24 is 264.
- Right, I hope that's right.
- We multiply the second equation by 9.
- So it's minus 63x plus 36y is equal to, let's see, 639.
- Big numbers.
- OK, now we add the two equations.
- Minus 12 minus 63 thats minus 75x-- these cancel out --equals
- 264, let's see what's 639 minus 264.
- See I do this in real time.
- I don't use some kind of solution manual or something.
- 13 and 5, 70.
- I don't know if I'm right, but we'll see.
- Since it's actually the negative 639, this is minus
- 375, and I know that seventy five goes into 300 4
- times, so x is equal to 5.
- 75 times 5 is 375.
- We just divided both sides by 75.
- So if x is 5 we just substitute it back into-- let's
- use this equation.
- So we get minus 3 times 5 minus 9y is equal to 66.
- We get minus 15 minus 9y equals 66.
- Minus 9y is equal to 81.
- And then we get y is equal to minus 9.
- So the answer is 5 comma minus 9.
- I think you're ready to do some systems of equations now.
- Have Fun.
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