[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.05,0:00:04.04,Default,,0000,0000,0000,,Welcome to the presentation on\Nsystems of linear equations.
Dialogue: 0,0:00:04.04,0:00:06.97,Default,,0000,0000,0000,,So let's get started and\Nsee what it's all about.
Dialogue: 0,0:00:06.97,0:00:10.11,Default,,0000,0000,0000,,So let's say I had\Ntwo equations now.
Dialogue: 0,0:00:10.11,0:00:15.74,Default,,0000,0000,0000,,The first equation let\Nme write it as 9x minus
Dialogue: 0,0:00:15.74,0:00:21.76,Default,,0000,0000,0000,,4y equals minus 78.
Dialogue: 0,0:00:21.76,0:00:28.95,Default,,0000,0000,0000,,And the second equation\NI will write as 4x plus
Dialogue: 0,0:00:28.95,0:00:33.39,Default,,0000,0000,0000,,y is equal to mine 18.
Dialogue: 0,0:00:33.39,0:00:35.41,Default,,0000,0000,0000,,Now what we're going to do now\Nis we're actually going to
Dialogue: 0,0:00:35.41,0:00:39.70,Default,,0000,0000,0000,,use both equations to\Nsolve for x and y.
Dialogue: 0,0:00:39.70,0:00:41.90,Default,,0000,0000,0000,,We already know that if you\Nhave one equation, it has one
Dialogue: 0,0:00:41.90,0:00:44.28,Default,,0000,0000,0000,,variable, it is very easy to\Nsolve for that one variable.
Dialogue: 0,0:00:44.28,0:00:45.79,Default,,0000,0000,0000,,But now we have to equations.
Dialogue: 0,0:00:45.79,0:00:47.34,Default,,0000,0000,0000,,You can almost view them\Nas two constraints.
Dialogue: 0,0:00:47.34,0:00:50.34,Default,,0000,0000,0000,,And we're going to solve\Nfor both variables.
Dialogue: 0,0:00:50.34,0:00:51.79,Default,,0000,0000,0000,,And you might be a\Nlittle confused.
Dialogue: 0,0:00:51.79,0:00:52.52,Default,,0000,0000,0000,,How does that work?
Dialogue: 0,0:00:52.52,0:00:54.91,Default,,0000,0000,0000,,Is it just magic that two\Nequations can solves
Dialogue: 0,0:00:54.91,0:00:55.90,Default,,0000,0000,0000,,for two variables?
Dialogue: 0,0:00:55.90,0:00:56.80,Default,,0000,0000,0000,,Well it's not.
Dialogue: 0,0:00:56.80,0:00:58.85,Default,,0000,0000,0000,,Because you can actually\Nrearranged each of these
Dialogue: 0,0:00:58.85,0:01:01.84,Default,,0000,0000,0000,,equations so that they\Nlook kind of in normal y
Dialogue: 0,0:01:01.84,0:01:03.70,Default,,0000,0000,0000,,equals mx plus b format.
Dialogue: 0,0:01:03.70,0:01:06.20,Default,,0000,0000,0000,,And I'm not going to draw these\Nactual two equations because I
Dialogue: 0,0:01:06.20,0:01:08.86,Default,,0000,0000,0000,,don't know what they look like,\Nbut if this was a coordinate
Dialogue: 0,0:01:08.86,0:01:11.62,Default,,0000,0000,0000,,axis-- and I don't know what\Nthat first line actually does
Dialogue: 0,0:01:11.62,0:01:14.01,Default,,0000,0000,0000,,look like, we could do another\Nmodel where we figured it out
Dialogue: 0,0:01:14.01,0:01:16.50,Default,,0000,0000,0000,,--but lets just say for sake of\Nargument, that first line all
Dialogue: 0,0:01:16.50,0:01:20.54,Default,,0000,0000,0000,,the x's and y's that satisfy 9x\Nminus 4y equals negative
Dialogue: 0,0:01:20.54,0:01:22.69,Default,,0000,0000,0000,,78, let's say it looks\Nsomething like that.
Dialogue: 0,0:01:22.69,0:01:26.40,Default,,0000,0000,0000,,And let's say all of the x's\Nand y's that satisfy that
Dialogue: 0,0:01:26.40,0:01:31.34,Default,,0000,0000,0000,,second equation, 4x plus y\Nequals negative 18, let's say
Dialogue: 0,0:01:31.34,0:01:34.68,Default,,0000,0000,0000,,that looks something like this.
Dialogue: 0,0:01:34.68,0:01:35.62,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:01:35.62,0:01:40.05,Default,,0000,0000,0000,,So, on the line is all of the\Nx's and y's that satisfy this
Dialogue: 0,0:01:40.05,0:01:42.56,Default,,0000,0000,0000,,equation, and on the green\Nline are all the x's and y's
Dialogue: 0,0:01:42.56,0:01:44.28,Default,,0000,0000,0000,,that satisfy this equation.
Dialogue: 0,0:01:44.28,0:01:48.17,Default,,0000,0000,0000,,But there's only one pair of\Nx and y's that satisfy both
Dialogue: 0,0:01:48.17,0:01:51.43,Default,,0000,0000,0000,,equations, and you can guess\Nwhere that is, that's
Dialogue: 0,0:01:51.43,0:01:52.56,Default,,0000,0000,0000,,right here right.
Dialogue: 0,0:01:52.56,0:01:57.66,Default,,0000,0000,0000,,Whatever that point is-- I'll\Ndo it in pink for emphasis.
Dialogue: 0,0:01:57.66,0:02:00.80,Default,,0000,0000,0000,,Whatever this point is,\Nnotice it's on both lines.
Dialogue: 0,0:02:00.80,0:02:05.26,Default,,0000,0000,0000,,So whatever x and y that is\Nwould be the solution to
Dialogue: 0,0:02:05.26,0:02:06.67,Default,,0000,0000,0000,,this system of equations.
Dialogue: 0,0:02:06.67,0:02:09.86,Default,,0000,0000,0000,,So let's actually figure\Nout how to do that.
Dialogue: 0,0:02:09.86,0:02:12.08,Default,,0000,0000,0000,,So what we want to do is\Neliminate a variable, because
Dialogue: 0,0:02:12.08,0:02:15.20,Default,,0000,0000,0000,,if you can eliminate a variable\Nthen we can just solve for
Dialogue: 0,0:02:15.20,0:02:16.43,Default,,0000,0000,0000,,the one that's left over.
Dialogue: 0,0:02:16.43,0:02:19.93,Default,,0000,0000,0000,,And the way to do that-- let's\Nsee, I want to eliminate, I
Dialogue: 0,0:02:19.93,0:02:22.21,Default,,0000,0000,0000,,feel like eliminating this y,\Nand I think you'll get
Dialogue: 0,0:02:22.21,0:02:24.63,Default,,0000,0000,0000,,an intuition for how we\Ncan do that later on.
Dialogue: 0,0:02:24.63,0:02:26.62,Default,,0000,0000,0000,,And the way I'm going to do\Nthat is I'm going to make
Dialogue: 0,0:02:26.62,0:02:29.25,Default,,0000,0000,0000,,it so that when I had this\Nto this, they cancel out.
Dialogue: 0,0:02:29.25,0:02:31.34,Default,,0000,0000,0000,,Well, they don't cancel out\Nright now, so I have to
Dialogue: 0,0:02:31.34,0:02:34.38,Default,,0000,0000,0000,,multiply this bottom equation\Nby 4, and I think it'll be
Dialogue: 0,0:02:34.38,0:02:35.52,Default,,0000,0000,0000,,obvious why I'm doing it.
Dialogue: 0,0:02:35.52,0:02:37.81,Default,,0000,0000,0000,,So let's multiply this\Nbottom equation by 4.
Dialogue: 0,0:02:37.81,0:02:50.82,Default,,0000,0000,0000,,And I get 16x plus 4y is equal\Nto 40 plus 32 minus 72.
Dialogue: 0,0:02:50.82,0:02:51.13,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:02:51.13,0:02:53.95,Default,,0000,0000,0000,,All I did is I multiplied\Nboth sides of the
Dialogue: 0,0:02:53.95,0:02:55.62,Default,,0000,0000,0000,,equation by 4, right?
Dialogue: 0,0:02:55.62,0:02:57.21,Default,,0000,0000,0000,,And you have to multiply\Nevery term because
Dialogue: 0,0:02:57.21,0:02:59.50,Default,,0000,0000,0000,,it's the distributive\Nproperty on both sides.
Dialogue: 0,0:02:59.50,0:03:01.05,Default,,0000,0000,0000,,Whatever you do to one side\Nyou have to do to the other.
Dialogue: 0,0:03:01.05,0:03:03.30,Default,,0000,0000,0000,,Let me rewrite top\Nequation again.
Dialogue: 0,0:03:03.30,0:03:05.23,Default,,0000,0000,0000,,And I'll write in the same\Ncolor so we can keep
Dialogue: 0,0:03:05.23,0:03:06.34,Default,,0000,0000,0000,,track of things.
Dialogue: 0,0:03:06.34,0:03:13.36,Default,,0000,0000,0000,,9x minus 4y is\Nequal to minus 78.
Dialogue: 0,0:03:13.36,0:03:18.58,Default,,0000,0000,0000,,OK, well now, if we were to add\Nthese two equations, when you
Dialogue: 0,0:03:18.58,0:03:20.43,Default,,0000,0000,0000,,add equations, you just add\Nthe left side and you
Dialogue: 0,0:03:20.43,0:03:22.27,Default,,0000,0000,0000,,add the right side.
Dialogue: 0,0:03:22.27,0:03:25.44,Default,,0000,0000,0000,,Well when you add, you\Nhave 16x plus 9x.
Dialogue: 0,0:03:25.44,0:03:28.59,Default,,0000,0000,0000,,Well that equals 25x.
Dialogue: 0,0:03:28.59,0:03:28.95,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:28.95,0:03:31.45,Default,,0000,0000,0000,,16 plus 9.
Dialogue: 0,0:03:31.45,0:03:34.91,Default,,0000,0000,0000,,4y minus 4, that just equals 0.
Dialogue: 0,0:03:34.91,0:03:43.68,Default,,0000,0000,0000,,So that's plus 0 equals, and\Nthen we have minus 72 minus 78.
Dialogue: 0,0:03:43.68,0:03:51.49,Default,,0000,0000,0000,,So, let's see that's minus\N150, minus 150, right?
Dialogue: 0,0:03:51.49,0:03:53.06,Default,,0000,0000,0000,,Just adding them all together.
Dialogue: 0,0:03:53.06,0:03:58.82,Default,,0000,0000,0000,,So we have 25x equals 150.
Dialogue: 0,0:03:58.82,0:04:03.42,Default,,0000,0000,0000,,Well, we could just divide both\Nsides by 25 or multiply both
Dialogue: 0,0:04:03.42,0:04:05.38,Default,,0000,0000,0000,,sides by 1/25, it's\Nthe same thing.
Dialogue: 0,0:04:05.38,0:04:08.47,Default,,0000,0000,0000,,And you get x equals--\Nthat's a negative 150
Dialogue: 0,0:04:08.47,0:04:11.50,Default,,0000,0000,0000,,--x equals minus 6.
Dialogue: 0,0:04:11.50,0:04:14.87,Default,,0000,0000,0000,,There we solved\Nthe x-coordinate.
Dialogue: 0,0:04:14.87,0:04:16.95,Default,,0000,0000,0000,,Now to solve the y-coordinate\Nwe can just use either one of
Dialogue: 0,0:04:16.95,0:04:18.50,Default,,0000,0000,0000,,these equations up at top.
Dialogue: 0,0:04:18.50,0:04:20.81,Default,,0000,0000,0000,,So let's use this one,\Nit seems a little bit,
Dialogue: 0,0:04:20.81,0:04:23.02,Default,,0000,0000,0000,,marginally simpler.
Dialogue: 0,0:04:23.02,0:04:26.09,Default,,0000,0000,0000,,So we just substitute the x\Nback in there and we get
Dialogue: 0,0:04:26.09,0:04:34.72,Default,,0000,0000,0000,,4 time minus 6 plus y\Nis equal to minus 18.
Dialogue: 0,0:04:34.72,0:04:35.73,Default,,0000,0000,0000,,Go up here.
Dialogue: 0,0:04:35.73,0:04:42.56,Default,,0000,0000,0000,,4 times minus 6 we get minus 24\Nplus y is equal to minus 18.
Dialogue: 0,0:04:42.56,0:04:47.41,Default,,0000,0000,0000,,And then get y is\Nequal to 24 minus 18.
Dialogue: 0,0:04:47.41,0:04:50.51,Default,,0000,0000,0000,,So y is equal to 6.
Dialogue: 0,0:04:50.51,0:04:54.10,Default,,0000,0000,0000,,So these two lines or these two\Nequations, you could even say,
Dialogue: 0,0:04:54.10,0:05:00.30,Default,,0000,0000,0000,,intersect at the point x is\Nm inus six and y is plus 6.
Dialogue: 0,0:05:00.30,0:05:02.52,Default,,0000,0000,0000,,So they actually intersect\Nsomeplace around here instead.
Dialogue: 0,0:05:02.52,0:05:05.64,Default,,0000,0000,0000,,I drew these, the line probably\Nlook something more like that.
Dialogue: 0,0:05:05.64,0:05:06.95,Default,,0000,0000,0000,,But that's pretty cool, no?
Dialogue: 0,0:05:06.95,0:05:11.83,Default,,0000,0000,0000,,We actually solved for two\Nvariables using two equations.
Dialogue: 0,0:05:11.83,0:05:12.64,Default,,0000,0000,0000,,Let's see how much time I have.
Dialogue: 0,0:05:12.64,0:05:14.47,Default,,0000,0000,0000,,I think we have enough time\Nto do another problem.
Dialogue: 0,0:05:14.47,0:05:20.20,Default,,0000,0000,0000,,\N105\N00:05:20,2 --> 00:05:23,02\NSo let's say I had the points--\Nand I'm going to write them in
Dialogue: 0,0:05:23.02,0:05:32.94,Default,,0000,0000,0000,,two different colors again\N--minus 7x minus 4y equals 9,
Dialogue: 0,0:05:32.94,0:05:39.15,Default,,0000,0000,0000,,and then the second equation is\Ngoing to be x plus
Dialogue: 0,0:05:39.15,0:05:42.46,Default,,0000,0000,0000,,2y is equal to 3.
Dialogue: 0,0:05:42.46,0:05:45.14,Default,,0000,0000,0000,,Now if I were doing this as\Nfast as possible, I'd probably
Dialogue: 0,0:05:45.14,0:05:47.99,Default,,0000,0000,0000,,multiply this equation times 7\Nand it would automatically
Dialogue: 0,0:05:47.99,0:05:49.02,Default,,0000,0000,0000,,cancel out.
Dialogue: 0,0:05:49.02,0:05:49.85,Default,,0000,0000,0000,,But that's easy way.
Dialogue: 0,0:05:49.85,0:05:51.29,Default,,0000,0000,0000,,I'm going to show you that\Nsometimes you might have to
Dialogue: 0,0:05:51.29,0:05:54.78,Default,,0000,0000,0000,,multiply both equations--\Nactually, not in this case.
Dialogue: 0,0:05:54.78,0:05:56.80,Default,,0000,0000,0000,,Actually let's just do it\Nthe fast way real fast.
Dialogue: 0,0:05:56.80,0:05:59.38,Default,,0000,0000,0000,,So let's multiply this\Nbottom equation by 7.
Dialogue: 0,0:05:59.38,0:06:00.83,Default,,0000,0000,0000,,And the whole reason why I want\Nto the, multiply it with 7,
Dialogue: 0,0:06:00.83,0:06:03.44,Default,,0000,0000,0000,,because I want this to\Ncancel out with this.
Dialogue: 0,0:06:03.44,0:06:10.15,Default,,0000,0000,0000,,If you multiply it by 7 you get\N7x plus 14y is equal to 21.
Dialogue: 0,0:06:10.15,0:06:12.93,Default,,0000,0000,0000,,Let's write that first\Nequation down again.
Dialogue: 0,0:06:12.93,0:06:19.06,Default,,0000,0000,0000,,Minus 7x minus 4y\Nis equal to 9.
Dialogue: 0,0:06:19.06,0:06:20.33,Default,,0000,0000,0000,,Now we just add.
Dialogue: 0,0:06:20.33,0:06:24.26,Default,,0000,0000,0000,,This is a positive 7x, it just\Nalways looks like a negative.
Dialogue: 0,0:06:24.26,0:06:25.90,Default,,0000,0000,0000,,OK, so that's 0.
Dialogue: 0,0:06:25.90,0:06:32.46,Default,,0000,0000,0000,,14 minus 4y plus 10y\Nis equal to 30.
Dialogue: 0,0:06:32.46,0:06:34.75,Default,,0000,0000,0000,,y is equal to 3.
Dialogue: 0,0:06:34.75,0:06:36.35,Default,,0000,0000,0000,,Now we just substitute back\Ninto either equation,
Dialogue: 0,0:06:36.35,0:06:37.98,Default,,0000,0000,0000,,lets do that one.
Dialogue: 0,0:06:37.98,0:06:42.11,Default,,0000,0000,0000,,x plus 2 times y, 2 times 3.
Dialogue: 0,0:06:42.11,0:06:43.88,Default,,0000,0000,0000,,x plus 6 equals 3.
Dialogue: 0,0:06:43.88,0:06:45.90,Default,,0000,0000,0000,,We get x equals negative 3.
Dialogue: 0,0:06:45.90,0:06:48.47,Default,,0000,0000,0000,,That one was super easy.
Dialogue: 0,0:06:48.47,0:06:49.55,Default,,0000,0000,0000,,The intercept.
Dialogue: 0,0:06:49.55,0:06:51.21,Default,,0000,0000,0000,,Hope I didn't do it to fast.
Dialogue: 0,0:06:51.21,0:06:54.43,Default,,0000,0000,0000,,Well, you can pause it and\Nwatch it again if you have.
Dialogue: 0,0:06:54.43,0:07:00.27,Default,,0000,0000,0000,,OK, so these two lines\Nintersect at the point
Dialogue: 0,0:07:00.27,0:07:03.18,Default,,0000,0000,0000,,negative 3 comma 3.
Dialogue: 0,0:07:03.18,0:07:04.25,Default,,0000,0000,0000,,Let's do one more.
Dialogue: 0,0:07:04.25,0:07:07.46,Default,,0000,0000,0000,,\N140\N00:07:07,456 --> 00:07:10,71\NHope this one's harder.
Dialogue: 0,0:07:10.71,0:07:11.51,Default,,0000,0000,0000,,I think it will.
Dialogue: 0,0:07:11.51,0:07:20.30,Default,,0000,0000,0000,,OK, negative 3x minus\N9y is equal to 66.
Dialogue: 0,0:07:20.30,0:07:27.20,Default,,0000,0000,0000,,We have minus 7x plus 4y\Nis equal to minus 71.
Dialogue: 0,0:07:27.20,0:07:28.37,Default,,0000,0000,0000,,So here it's not obvious.
Dialogue: 0,0:07:28.37,0:07:31.54,Default,,0000,0000,0000,,What we have to do is, let's\Nsay we want to cancel
Dialogue: 0,0:07:31.54,0:07:33.98,Default,,0000,0000,0000,,out the y's first.
Dialogue: 0,0:07:33.98,0:07:36.50,Default,,0000,0000,0000,,What we do is we try to make\Nboth of them equal to the least
Dialogue: 0,0:07:36.50,0:07:38.66,Default,,0000,0000,0000,,common multiple of 9 and 4.
Dialogue: 0,0:07:38.66,0:07:43.34,Default,,0000,0000,0000,,So, if we multiply the top\Nequation by 4 we get--
Dialogue: 0,0:07:43.34,0:07:44.52,Default,,0000,0000,0000,,I'll do it right here.
Dialogue: 0,0:07:44.52,0:07:45.87,Default,,0000,0000,0000,,Let's multiply it by 4.
Dialogue: 0,0:07:45.87,0:07:47.96,Default,,0000,0000,0000,,Times 4.
Dialogue: 0,0:07:47.96,0:07:59.20,Default,,0000,0000,0000,,We'll get minus 12x minus\N36y is equal to 4 times
Dialogue: 0,0:07:59.20,0:08:05.40,Default,,0000,0000,0000,,240 plus 24 is 264.
Dialogue: 0,0:08:05.40,0:08:06.93,Default,,0000,0000,0000,,Right, I hope that's right.
Dialogue: 0,0:08:06.93,0:08:09.22,Default,,0000,0000,0000,,We multiply the second\Nequation by 9.
Dialogue: 0,0:08:09.22,0:08:25.42,Default,,0000,0000,0000,,So it's minus 63x plus 36y is\Nequal to, let's see, 639.
Dialogue: 0,0:08:25.42,0:08:26.03,Default,,0000,0000,0000,,Big numbers.
Dialogue: 0,0:08:26.03,0:08:29.35,Default,,0000,0000,0000,,639.
Dialogue: 0,0:08:29.35,0:08:31.54,Default,,0000,0000,0000,,OK, now we add the\Ntwo equations.
Dialogue: 0,0:08:31.54,0:08:43.57,Default,,0000,0000,0000,,Minus 12 minus 63 thats minus\N75x-- these cancel out --equals
Dialogue: 0,0:08:43.57,0:08:50.13,Default,,0000,0000,0000,,264, let's see what's\N639 minus 264.
Dialogue: 0,0:08:50.13,0:08:51.16,Default,,0000,0000,0000,,See I do this in real time.
Dialogue: 0,0:08:51.16,0:08:55.10,Default,,0000,0000,0000,,I don't use some kind of\Nsolution manual or something.
Dialogue: 0,0:08:55.10,0:08:59.71,Default,,0000,0000,0000,,13 and 5, 70.
Dialogue: 0,0:08:59.71,0:09:02.26,Default,,0000,0000,0000,,I don't know if I'm\Nright, but we'll see.
Dialogue: 0,0:09:02.26,0:09:06.36,Default,,0000,0000,0000,,Since it's actually the\Nnegative 639, this is minus
Dialogue: 0,0:09:06.36,0:09:12.44,Default,,0000,0000,0000,,375, and I know that seventy\Nfive goes into 300 4
Dialogue: 0,0:09:12.44,0:09:16.45,Default,,0000,0000,0000,,times, so x is equal to 5.
Dialogue: 0,0:09:16.45,0:09:19.52,Default,,0000,0000,0000,,75 times 5 is 375.
Dialogue: 0,0:09:19.52,0:09:22.46,Default,,0000,0000,0000,,We just divided\Nboth sides by 75.
Dialogue: 0,0:09:22.46,0:09:25.37,Default,,0000,0000,0000,,So if x is 5 we just substitute\Nit back into-- let's
Dialogue: 0,0:09:25.37,0:09:27.89,Default,,0000,0000,0000,,use this equation.
Dialogue: 0,0:09:27.89,0:09:36.38,Default,,0000,0000,0000,,So we get minus 3 times 5\Nminus 9y is equal to 66.
Dialogue: 0,0:09:36.38,0:09:41.92,Default,,0000,0000,0000,,We get minus 15\Nminus 9y equals 66.
Dialogue: 0,0:09:41.92,0:09:45.88,Default,,0000,0000,0000,,Minus 9y is equal to 81.
Dialogue: 0,0:09:45.88,0:09:49.84,Default,,0000,0000,0000,,And then we get y is\Nequal to minus 9.
Dialogue: 0,0:09:49.84,0:09:53.53,Default,,0000,0000,0000,,So the answer is\N5 comma minus 9.
Dialogue: 0,0:09:53.53,0:09:55.53,Default,,0000,0000,0000,,I think you're ready to do some\Nsystems of equations now.
Dialogue: 0,0:09:55.53,0:09:57.09,Default,,0000,0000,0000,,Have Fun.