Simultaneous Equations Three Variables Using Elimination - Math lesson - Free Educational videos for Students in K-12

Simultaneous Equations Three Variables Using Elimination - Math lesson - By tecmath

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00:0-1	today . Welcome to take Math Channel . I'm josh
00:02	in this video . We're going to be using the
00:04	elimination method to solve simultaneous equations that involve three variables
00:08	. So ones like you can see here , we
00:10	have three equations and they have three variables X ,
00:14	Y , and Z . And the way we're going
00:16	to do this is as follows . We're going to
00:19	By combining reduce these equations down to two equations that
00:23	involve two variables . And that will then allow us
00:26	to use the elimination method on two variable two equations
00:29	to reduce down to one variable and go back from
00:32	there . So as you see this is how this
00:34	works . So the way that we're going to do
00:37	this is as follows , the first thing we're going
00:38	to do is we are going to combine two of
00:41	these equations here . So the ones I'm going to
00:43	combine this equation , one equation to . And you
00:46	can see that by doing this . If you look
00:47	at the Z variable , we have negative said here
00:50	and positive said here , these guys are gonna cancel
00:52	each other out and we're gonna be left with an
00:54	equation with only two variables the X and Y .
00:56	So let's do that . So he would have combined
00:59	these guys three X And two x . gives us
01:03	five x and positive two Y negative three Y .
01:08	Gives us negative y , Z's counsel each other out
01:11	negatives and positives and cancel each other out . And
01:14	this is equal to 11 Plus seven , which is
01:17	equal to 18 . So we have our first equation
01:19	which has only two variables X&y . The next thing
01:23	we're going to do is we're going to combine two
01:24	other equations but the one we have to combine ,
01:27	it has to have this third equation in it in
01:29	order for this to work . So we're going to
01:31	combine equation two and equation three here , as you
01:35	can see if we consider the zen variable , you
01:37	can see that this is positive , said and this
01:39	is negative Tuesday . So what we're going to have
01:41	to do is we are going to have to multiply
01:43	this entire equation by two . So it wouldn't do
01:46	that . This is what we would get . So
01:50	we would get to X times two which is equal
01:53	to four , X minus three Y times two which
01:57	is minus six Y . And positive to Z .
02:02	This is going to be equal to seven times two
02:05	which is 14 . Cool . So equations now that
02:09	we're actually going to be combining is the modified version
02:13	of this second equation that we have down here .
02:15	So let's go through and let's do that . So
02:18	what do we get when we do that ? I'm
02:20	going to write the answer underneath . So five x
02:24	Plus forex gives us nine x positive way by the
02:29	six Y gives us negative five Y negative 20 and
02:34	positive to Z . Well they cancel each other out
02:36	and this is going to be equal to 12 plus
02:39	14 which is 26 . So now as you can
02:42	see we have two equations here , the ones that
02:44	are in blue and both of them have really have
02:47	two variables . They have the X . Variable and
02:49	the Y variable as you can see . So now
02:51	what we can do is we can combine these two
02:53	equations and what we can do is reduce down to
02:56	only one variable . So let's go through and do
02:59	that . So if we have a look at our
03:00	two equations , I'm going to eliminate the y variable
03:04	. Okay , leave only leaving us with the Xia
03:06	. So the way that I'm going to do that
03:08	, if you have a look at that we have
03:09	negative fire by here and negative Y here . So
03:12	I multiplies entire equation by negative five . I'll end
03:16	up with a positive five Y . Which will cancel
03:18	it down here . So I'm going to give the
03:20	result of that down here . So let's do that
03:22	. What do we have ? We have five X
03:25	times negative five is going to be negative 25 six
03:30	minus Y . Times negative five is positive five .
03:34	Y . We also have 18 times negative five ,
03:38	which is going to give us -90 . Cool .
03:42	All right now , that means we can go through
03:44	and eliminate because as you can see these guys are
03:47	going to cancel each other out . What do we
03:49	end up with ? We end up with nine x
03:51	minus 25 X . Which is equal to negative 16
03:55	X . And this is equal to 26 minus 90
03:59	which is equal to negative 64 . Okay , easy
04:03	to solve 16 . Well , negative 16 goes into
04:06	both of them , so we end up with X
04:08	equals four and that's the very very first variable whether
04:12	we managed to solve their . Now what we do
04:14	is we're going to substitute this into one of our
04:17	equations that has two variables , the X and the
04:19	Y . We're going to find out why is and
04:21	then we're gonna use our answers there and substitute into
04:23	one of these equations and then we'll have all our
04:25	answers . So let's go through and do that .
04:27	So first off let's substitute X equals four into one
04:32	of these equations . Let's put it into this one
04:34	up here . Okay , so we have x equals
04:36	four . This is going to be five times four
04:39	minus Y equals 18 . Cool , so what we
04:44	end up with , we end up with 20 minus
04:46	Y equals 18 and therefore when we saw this we
04:50	take 20 off both sides . We end up with
04:53	negative too . So why is equal to two ?
04:56	20-2 is equal to 18 ? So I'm gonna actually
05:00	put that in a different color . We have a
05:02	second variable here . That is why is equal to
05:06	two . All right , cool . Now , what
05:09	we do is we have X equals four . We
05:11	have vehicles to we substitute these into one of our
05:15	original equations . We're going to find out what's N
05:17	. Equals So let's go through and do that .
05:19	I'm just gonna ride out this entire equation again .
05:21	So we have three X Plus to why ? This
05:24	is equation one -Z equals 11 . All right ,
05:29	So what do we do we work this one out
05:31	of three ? Ex well , X is equal to
05:33	four , So 34 12 plus two , I two
05:38	times two is equal to four miners . E equals
05:42	11 . Cool . 12 plus four is equal to
05:46	16 minus Z equals 11 . I reckon you'll be
05:49	able to see what's N equals 2 . 16 minus
05:52	five is equal to 11 . So Z is equal
05:56	to five . We have our three variables here we
05:58	have X is equal to four , we have Y
06:00	is equal to two and Z is equal to five
06:04	. You go through and sub these in just to
06:06	check you . Ok Two times four is eight minus
06:09	six . There to minus six is two plus 57
06:14	So it's correct . Their 20 plus two's 20 to
06:18	minus two times five which is 10 . Uh 20
06:22	to -10 is equal to 12 . So go through
06:25	check , make sure it's okay . And that's how
06:28	you solve that . We reduced from three equations with
06:30	three variables . Standard two equations with two variables and
06:33	then down to one equation , one variable . Anyway
06:36	. That's how you solve simultaneous equations using three variables
06:40	and the elimination method . Thank you for watching .
06:42	We'll see you next time . Bye .

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