0:00:01.630,0:00:06.274
The purpose of this video is to[br]look at the solution of

0:00:06.274,0:00:07.048
elementary simultaneous

0:00:07.048,0:00:12.884
equations. Before we do that,[br]let's just have a look at a

0:00:12.884,0:00:13.732
relatively straightforward

0:00:13.732,0:00:20.162
single equation. Equation we're[br]going to look at 2X minus

0:00:20.162,0:00:21.899
Y equals 3.

0:00:22.900,0:00:26.651
This is a linear equation. It's[br]a linear equation because there

0:00:26.651,0:00:31.766
are no terms in it that are X[br]squared, Y squared or X times by

0:00:31.766,0:00:36.540
Y or indeed ex cubes. The only[br]terms we've got a terms in X

0:00:36.540,0:00:38.586
terms in Y and some numbers.

0:00:39.700,0:00:42.775
So this represents a linear

0:00:42.775,0:00:48.260
equation. We can rearrange it so[br]that it says why equal

0:00:48.260,0:00:53.330
something, so let's just do[br]that. We can add Y to each side

0:00:53.330,0:00:55.280
so that we get 2X.

0:00:55.960,0:01:01.028
Equals 3 plus Yi. Did say add[br]why two each side and you might

0:01:01.028,0:01:05.010
have wondered what happened[br]here. Well, if I've got minus Y

0:01:05.010,0:01:10.078
and I add why to it, I end up[br]with no wise at all.

0:01:11.400,0:01:18.414
Here we've got two X equals 3[br]plus Y, so let's take the three

0:01:18.414,0:01:22.923
away from each side. 2X minus[br]three equals Y.

0:01:23.700,0:01:29.556
So there I've got a nice[br]expression for why. If I take

0:01:29.556,0:01:36.876
any value of X. Let's say I take[br]X equals 1, then why will be

0:01:36.876,0:01:44.196
equal to two times by 1 - 3,[br]which gives us minus one. So for

0:01:44.196,0:01:51.028
this value of XI, get that value[br]of Yi can take another value of

0:01:51.028,0:01:53.956
XX equals 2, Y will equal.

0:01:54.020,0:01:58.340
2 * 2 - 3, which is[br]plus one.

0:02:00.720,0:02:06.245
Another value of XX equals[br]0 Y equals 2 times by 0 -

0:02:06.245,0:02:11.770
3 will 2 * 0 is 0 and that[br]gives me minus three.

0:02:13.280,0:02:17.924
So for every value of XI can[br]generate a value of Y.

0:02:19.260,0:02:24.083
I can plot these as point[br]so I can plot this as the

0:02:24.083,0:02:29.648
.1 - 1 and I can plot this[br]one on a graph as the .21

0:02:29.648,0:02:34.100
and this one on a graph as[br]the point nort minus three.

0:02:34.100,0:02:36.326
So let's just set that up.

0:02:39.700,0:02:40.828
Pair of axes.

0:02:43.560,0:02:47.733
Let's mark the values of X that[br]we've been having a look at.

0:02:49.610,0:02:53.378
So that was X on there and why

0:02:53.378,0:02:57.372
there? And let's put on the[br]values of why that we got.

0:02:59.650,0:03:06.074
When X was zero hour value[br]of why was minus three?

0:03:06.910,0:03:08.770
So that's. There.

0:03:10.500,0:03:13.979
When X was one hour value was

0:03:13.979,0:03:19.977
minus one. And when X was[br]two hour value was one.

0:03:21.430,0:03:25.814
Those three points lie on[br]a straight line.

0:03:27.780,0:03:32.900
Y equals 2X minus three, and[br]that's another reason for

0:03:32.900,0:03:38.532
calling this a linear equation.[br]It gives us a straight line.

0:03:39.650,0:03:40.360
OK.

0:03:42.190,0:03:49.613
You've got that one. Y equals 2X[br]minus three. Supposing we take a

0:03:49.613,0:03:57.036
second one 3X plus two Y equals[br]8, a second linear equation, and

0:03:57.036,0:03:59.891
supposing we say these two.

0:04:00.920,0:04:03.386
Are true at the same time.

0:04:05.320,0:04:06.480
What does that mean?

0:04:07.060,0:04:11.176
Well, we can plot this as a[br]straight line. Again, it's a

0:04:11.176,0:04:15.292
linear equation, so it's going[br]to give us a straight line. Now

0:04:15.292,0:04:20.094
I don't want to have to workout[br]lots of points for this, So what

0:04:20.094,0:04:24.896
I'm going to do is just sketch[br]it in quickly on the graph. I'm

0:04:24.896,0:04:27.297
going to say when X is 0.

0:04:27.910,0:04:32.875
And cover up the Exterm 2 Y is[br]equal to 8, so why must be

0:04:32.875,0:04:36.516
equal to four which is going[br]to be up there somewhere?

0:04:38.030,0:04:45.204
And when? Why is 03 X is equal[br]to 8 and so X is 8 / 3

0:04:45.204,0:04:50.268
which gives us 2 and 2/3. So[br]somewhere about their two 2/3

0:04:50.268,0:04:55.754
and we know it's a straight[br]line, so we can get that by

0:04:55.754,0:04:57.020
joining up there.

0:04:58.400,0:05:05.480
This is the equation 3X plus two[br]Y equals 8. So what does it mean

0:05:05.480,0:05:12.088
for these two to be true at the[br]same time? Well, it must mean

0:05:12.088,0:05:13.976
it's this point here.

0:05:14.560,0:05:19.756
Where the two lines cross. So[br]when we solve a pair of

0:05:19.756,0:05:23.220
simultaneous equations, what[br]we're actually looking for is

0:05:23.220,0:05:25.818
the intersection of two[br]straight lines.

0:05:26.850,0:05:32.430
Of course, it could happen that[br]we have one line like that.

0:05:33.400,0:05:34.828
And apparel line.

0:05:35.970,0:05:37.218
They would never meet.

0:05:38.170,0:05:42.190
And one of the examples that[br]we're going to be looking at

0:05:42.190,0:05:45.540
later will show what happens[br]in terms of the arithmetic

0:05:45.540,0:05:49.225
when we have this particular[br]case. But for now, let's go

0:05:49.225,0:05:52.910
back and think about these[br]two. How can we handle these

0:05:52.910,0:05:56.595
two algebraically so that we[br]don't have to draw graphs? We

0:05:56.595,0:05:59.610
don't have to rely on[br]sketching, we can calculate

0:05:59.610,0:06:02.960
which is so much easier in[br]most cases that actually

0:06:02.960,0:06:03.965
drawing a graph.

0:06:05.250,0:06:08.868
So let's take these two[br]equations.

0:06:12.850,0:06:18.202
And we're going to look at two[br]methods of solution, so I'm

0:06:18.202,0:06:23.108
going to look at method one.[br]Now, let's begin with the

0:06:23.108,0:06:28.906
original equation that we had[br]two X minus. Y is equal to three

0:06:28.906,0:06:35.596
and then the one that we put[br]with it 3X plus two Y equals 8.

0:06:37.360,0:06:41.728
Our first method of solution,[br]well, one of the things to do is

0:06:41.728,0:06:46.096
to do what we did in the very[br]first case with this and

0:06:46.096,0:06:48.784
Rearrange one of these[br]equations. It doesn't matter

0:06:48.784,0:06:50.800
which one, but we'll take this

0:06:50.800,0:06:54.760
one. So that we get Y

0:06:54.760,0:07:01.424
equals. And we know what the[br]result for that one is. It's Y

0:07:01.424,0:07:03.152
equals 2X minus three.

0:07:05.230,0:07:07.518
So that's equation one.

0:07:08.170,0:07:15.720
That's equation. Two, so this[br]is now, let's call it equation 3

0:07:15.720,0:07:18.870
and we got it by rearranging.

0:07:21.150,0:07:21.940
1.

0:07:23.610,0:07:29.325
What we're going to do with this[br]is if these two have to be true

0:07:29.325,0:07:33.516
at the same time, then this[br]relationship must be true in

0:07:33.516,0:07:37.326
this equation, so we can[br]substitute it in, so let's.

0:07:38.350,0:07:40.990
Substitute 3

0:07:42.460,0:07:48.816
until two so we[br]have 3X plus.

0:07:50.290,0:07:58.222
Two times Y. But why is[br]2X minus three that's equal to

0:07:58.222,0:08:02.510
8? And you can see that what[br]we've done is we've reduced.

0:08:03.190,0:08:08.480
This. To this equation giving us[br]a single equation in one

0:08:08.480,0:08:12.583
unknown, which is a simple[br]linear equation and we can solve

0:08:12.583,0:08:19.922
it. Multiply out the brackets 3X[br]plus 4X minus 6 equals 8.

0:08:21.150,0:08:26.694
Gather the excess together. 7X[br]minus 6 equals 8.

0:08:27.470,0:08:33.454
At the six to each side, Seven X[br]equals 14, and so X must be 2.

0:08:35.020,0:08:37.090
That's only given as one value.

0:08:38.060,0:08:42.870
We need a value of Y, but up[br]here we've got an expression

0:08:42.870,0:08:48.050
which says Y equals and if we[br]take the value of X that we've

0:08:48.050,0:08:49.530
got and substituted in.

0:08:50.420,0:08:55.033
Therefore, why[br]will be equal to 2

0:08:55.033,0:08:58.987
* 2 - 3 gives US1?

0:09:00.600,0:09:05.472
And so we've got a solution X[br]equals 2, Y equals 1.

0:09:06.910,0:09:08.280
Are we sure it's right?

0:09:09.040,0:09:13.891
Well, we used this equation[br]which came from equation one to

0:09:13.891,0:09:16.096
generate the value of Y.

0:09:16.740,0:09:22.270
So if we take the values of X&Y[br]and put them back into here,

0:09:22.270,0:09:27.010
they should work, should give us[br]the right answer. So let's try

0:09:27.010,0:09:28.590
that. X is 2.

0:09:29.350,0:09:37.036
3 times by two is 6 plus, Y is[br]1 two times by one is 2 six and

0:09:37.036,0:09:42.160
two gives us eight. Yes, this[br]works. This is a solution of

0:09:42.160,0:09:47.711
that equation and of that one.[br]So this is our answer to the

0:09:47.711,0:09:49.419
pair of simultaneous equations.

0:09:50.320,0:09:54.489
Let's have a look at another one[br]using this particular method.

0:09:57.420,0:10:00.038
The example we're going to use[br]is going to be said.

0:10:00.110,0:10:07.195
Open X. +2[br]Y equals 47 and

0:10:07.195,0:10:13.922
five X minus four[br]Y equals 1.

0:10:15.930,0:10:21.290
Now. We need to make a choice.[br]We need to choose one of these

0:10:21.290,0:10:27.503
two equations. And Rearrange it[br]so that it says Y equals or if

0:10:27.503,0:10:29.235
we want X equals.

0:10:31.190,0:10:35.740
The choice is entirely ours and[br]we have to make the choice based

0:10:35.740,0:10:40.290
upon what we feel will be the[br]simplest and looking at a pair

0:10:40.290,0:10:44.140
of equations like this often[br]difficult to know which is the

0:10:44.140,0:10:47.640
simplest. Well, let's pick at[br]random. Let's choose this one

0:10:47.640,0:10:51.490
and let's Rearrange Equation[br]too. So we'll start by getting X

0:10:51.490,0:10:56.740
equals this time. So we say 5X[br]is equal to 1 and I'm going to

0:10:56.740,0:10:59.540
add 4 Y to each side plus 4Y.

0:11:00.580,0:11:04.756
Now I'm going to divide[br]throughout by the five so that I

0:11:04.756,0:11:10.494
have. X on its own. Now I've got[br]to divide everything by 5.

0:11:11.130,0:11:15.066
Everything so I had to put[br]that line there to show that

0:11:15.066,0:11:19.330
I'm dividing the one and the[br]four Y. So this is a fraction.

0:11:19.330,0:11:24.250
I'm sure you can tell this is[br]not going to be as easy as the

0:11:24.250,0:11:27.530
previous question was. In[br]fact, it's going to be quite

0:11:27.530,0:11:31.138
difficult because I have to[br]take this now and because it

0:11:31.138,0:11:32.450
came from equation too.

0:11:37.660,0:11:42.626
I'm going to have to take it and[br]substitute it back into equation

0:11:42.626,0:11:47.592
one, and this isn't looking very[br]pretty, so let's give it a try

0:11:47.592,0:11:49.460
sub. 3.

0:11:51.180,0:11:59.002
Until one. So I[br]have 7X but X is this

0:11:59.002,0:12:06.086
lump of algebra here 1 +[br]4 Y all over 5.

0:12:06.730,0:12:10.501
+2 Y equals

0:12:10.501,0:12:15.604
47. I can see this is[br]becoming quite horrific.

0:12:16.620,0:12:20.547
Multiply throughout by 5 why?[br]Because we're dividing by 5. We

0:12:20.547,0:12:25.545
want to get rid of the fraction.[br]The way to do that is to

0:12:25.545,0:12:30.186
multiply everything by 5 and it[br]has to be everything. So if we

0:12:30.186,0:12:34.827
multiply that by 5 because we're[br]dividing by 5, it's as though we

0:12:34.827,0:12:41.253
actually do nothing to the 1 + 4[br]Y. That leaves a 7 * 1 + 4 Y.

0:12:41.910,0:12:47.678
We need five times that that's[br]ten Y and we have to have five

0:12:47.678,0:12:51.386
times that remember, an[br]equation is a balance. What

0:12:51.386,0:12:57.154
you do to one side of the[br]balance you have to do to the

0:12:57.154,0:13:00.450
other. If you don't, it's[br]unbalanced. So we're

0:13:00.450,0:13:05.394
multiplying everything by 5.[br]So 5 * 47 five 735, five falls

0:13:05.394,0:13:06.630
of 22135 altogether.

0:13:07.770,0:13:15.386
Now we need to multiply out the[br]brackets 7 + 28 Y plus 10

0:13:15.386,0:13:19.194
Y equals 235. So we take this

0:13:19.194,0:13:23.692
equation. Write it down again so[br]that we can see it clearly.

0:13:30.920,0:13:35.240
Now we can gather these two[br]together gives us 38Y.

0:13:36.310,0:13:41.920
And we can take Seven away[br]from each side, which will

0:13:41.920,0:13:43.450
give us 228.

0:13:44.910,0:13:49.827
Exactly big numbers coming in[br]here 228 / 38 'cause we're

0:13:49.827,0:13:54.744
looking for the number which[br]when we multiplied by 38 will

0:13:54.744,0:13:57.873
give us 228 and that's going to

0:13:57.873,0:14:03.020
be 6. So we've established Y is[br]equal to 6.

0:14:03.950,0:14:09.202
Having done that, we can take it[br]and we can substitute it back

0:14:09.202,0:14:14.454
into the equation that we first[br]had for X. So remember that for

0:14:14.454,0:14:17.282
that we had X was equal to.

0:14:18.100,0:14:25.300
And what we had for that was 1[br]+ 4 Y all over five. We

0:14:25.300,0:14:32.020
substitute in the six, so we[br]have 1 + 24 or over 5 and

0:14:32.020,0:14:39.220
quickly we can see that's 25 /[br]5. So we have X equals 5. So

0:14:39.220,0:14:44.980
again we've got our pair of[br]values. Our answer to the pair

0:14:44.980,0:14:48.820
of simultaneous equations. We[br]haven't checked it though.

0:14:49.390,0:14:53.845
Now remember that this came from[br]the second equation, so really

0:14:53.845,0:14:59.515
to check it we've got to go back[br]to the very first equation that

0:14:59.515,0:15:05.185
we had written down that one. If[br]you remember was Seven X +2, Y

0:15:05.185,0:15:06.805
is equal to 47.

0:15:07.880,0:15:15.344
So let's just check 7 *[br]5. That gives us 35 +

0:15:15.344,0:15:17.210
2 * 6.

0:15:17.900,0:15:23.340
That gives us 12, so we 35 + 12[br]equals 47. And yes, that is what

0:15:23.340,0:15:28.100
we wanted, so we now know that[br]this is correct, but I just stop

0:15:28.100,0:15:29.460
and think about it.

0:15:30.040,0:15:34.876
We got all those fractions to[br]work with. We got this lump of

0:15:34.876,0:15:39.712
algebra to carry around with us.[br]Is there not an easier way of

0:15:39.712,0:15:44.820
doing these? Yes there is. It's[br]useful to have seen the method

0:15:44.820,0:15:49.760
that we have got simply because[br]we will need it again when we

0:15:49.760,0:15:53.180
look at the second video of[br]simultaneous equations, but.

0:15:54.030,0:15:58.566
That is a simple way of handling[br]these, so let's go on now and

0:15:58.566,0:16:00.510
have a look at method 2.

0:16:09.300,0:16:12.850
Now this method is sometimes[br]called elimination and we can

0:16:12.850,0:16:17.465
see why it gets that name and[br]this is the method that you

0:16:17.465,0:16:21.370
really do need to practice and[br]become accustomed to. So let's

0:16:21.370,0:16:24.920
start with the same equations[br]that we had last time.

0:16:26.000,0:16:26.980
And see.

0:16:28.540,0:16:32.630
How it works and how much easier[br]it actually is?

0:16:33.780,0:16:37.316
OK method of elimination. What[br]do we do?

0:16:38.180,0:16:42.428
What we do is we seek[br]to make the

0:16:42.428,0:16:45.260
coefficients in front[br]of the wise.

0:16:46.410,0:16:48.570
Or in front of the axes.

0:16:49.430,0:16:50.210
The same.

0:16:51.650,0:16:56.094
Once we've gotten the same,[br]then we can either add the

0:16:56.094,0:16:59.326
two equations together or[br]subtract them according to

0:16:59.326,0:17:01.346
the signs that are there.

0:17:02.780,0:17:08.300
By doing that, we will get rid[br]of that particular unknown, the

0:17:08.300,0:17:10.140
one that we chose.

0:17:10.750,0:17:14.306
To make the coefficients[br]numerically the same.

0:17:15.720,0:17:18.886
So. This one what would we do?

0:17:19.560,0:17:22.731
Well, if we look at this and

0:17:22.731,0:17:28.910
this. Here we have two Y and[br]here we have minus four Y.

0:17:29.540,0:17:34.398
So if I were to double that, I'd[br]have four. Why there? And it's

0:17:34.398,0:17:38.909
plus four Y Ana minus four Y[br]there, and that seems are pretty

0:17:38.909,0:17:43.420
good thing to do, because then[br]they're both for Y. One of them

0:17:43.420,0:17:47.931
is plus and one of them is[br]minus. And if I add them

0:17:47.931,0:17:51.748
together they will disappear. So[br]let me just number the equations

0:17:51.748,0:17:52.789
one and two.

0:17:53.650,0:17:59.050
And then I can keep a record of[br]what I'm doing. So I'm going to

0:17:59.050,0:18:03.010
multiply the first equation by[br]two and that's going to lead

0:18:03.010,0:18:05.170
Maine to a new equation 3.

0:18:05.780,0:18:07.000
So let's do that.

0:18:07.820,0:18:14.792
2 * 7 X is 14X[br]plus two times, that is 4

0:18:14.792,0:18:21.183
Y equals 2 times that, and[br]2 * 47 is 94.

0:18:22.190,0:18:27.949
Now equation two. I'm leaving as[br]it is not going to touch it.

0:18:30.510,0:18:32.620
Now I've got two equations.

0:18:34.110,0:18:41.194
This is plus four Y and this[br]is minus four Y. So if I

0:18:41.194,0:18:47.266
add the two equations together,[br]what happens? I get 14X plus 5X.

0:18:47.266,0:18:48.784
That's 19 X.

0:18:50.880,0:18:56.775
No whys. 'cause I've plus four Y[br]add it to minus four Y know wise

0:18:56.775,0:18:57.954
at all equals.

0:18:58.820,0:18:59.850
95

0:19:01.010,0:19:06.386
and so X is 95 over 19, which[br]gives me 5 which if you

0:19:06.386,0:19:10.226
remember, is the answer we have[br]to the last question.

0:19:11.820,0:19:15.285
Now we need to take this and[br]substitute it back. Doesn't

0:19:15.285,0:19:18.435
matter which equation we choose[br]to substitute it back into.

0:19:18.435,0:19:19.695
Let's take this one.

0:19:20.350,0:19:27.562
X is 5, so five times[br]by 5 - 4 Y equals

0:19:27.562,0:19:28.163
1.

0:19:29.200,0:19:35.090
And so we have 25 -[br]4 Y equals 1.

0:19:37.200,0:19:39.930
Take the four way over[br]to that side by adding

0:19:39.930,0:19:41.295
four Y to each side.

0:19:43.260,0:19:49.116
So that will give us 25 is equal[br]to four Y plus one. Take the one

0:19:49.116,0:19:54.972
away from its side, 24 is 4 Y[br]and so why is equal to 6 and

0:19:54.972,0:19:58.998
we've got exactly the same[br]answer as we had before. And

0:19:58.998,0:20:00.096
let's just look.

0:20:01.400,0:20:05.426
How much simpler that is? How[br]much quicker that answer came

0:20:05.426,0:20:09.818
out. One thing to notice. Well,[br]two things in actual fact. First

0:20:09.818,0:20:14.210
of all, I try to keep the equal[br]signs underneath each other.

0:20:15.060,0:20:19.246
This is not only makes it look[br]neat, it enables you to see what

0:20:19.246,0:20:20.442
it is you're doing.

0:20:21.540,0:20:25.082
Keep the equations together so[br]the setting out of this work

0:20:25.082,0:20:27.336
actually helps you to be able to

0:20:27.336,0:20:32.783
check it. Second thing to notice[br]is down this side. I've kept a

0:20:32.783,0:20:36.886
record of exactly what I've done[br]multiplying the equation by two,

0:20:36.886,0:20:39.870
adding the two equations[br]together. That's very helpful

0:20:39.870,0:20:45.092
when you want to check your[br]work. What did I do? How did I

0:20:45.092,0:20:48.822
actually work this out? By[br]having this record down the

0:20:48.822,0:20:53.671
side, you don't have to work it[br]out again. You can see exactly

0:20:53.671,0:20:58.520
what it is that you did. Now[br]let's take a third example and

0:20:58.520,0:21:03.218
again. Will solve it by means of[br]the method of elimination. Just

0:21:03.218,0:21:08.650
so we've got a second example of[br]that method to look at three X

0:21:08.650,0:21:10.978
+7 Y is 27 and 5X.

0:21:12.620,0:21:14.968
+2 Y is 60.

0:21:16.430,0:21:20.234
OK, we've got a choice to make.[br]We can make either the

0:21:20.234,0:21:23.721
coefficients in front of the axe[br]numerically the same, or the

0:21:23.721,0:21:27.525
coefficients in front of the[br]wise. Well, in order to do that,

0:21:27.525,0:21:31.012
I'd have to multiply the Y.[br]Certainly have to multiply this

0:21:31.012,0:21:35.450
equation by two to give me 14[br]there and this one by 7:00 to

0:21:35.450,0:21:36.718
give me 14 there.

0:21:38.310,0:21:39.934
How do I make that choice? Well?

0:21:40.990,0:21:45.742
Fairly clearly 2 times by 7 is[br]14, so 1 by 1, one by the other.

0:21:46.660,0:21:49.918
But I don't really like[br]multiplying by 7 difficult

0:21:49.918,0:21:54.262
number. I prefer to multiply by[br]three and five, so my choices

0:21:54.262,0:21:58.244
actually governed by how well I[br]think I can handle the

0:21:58.244,0:22:02.588
arithmetic. So let's multiply[br]this one by 5 and this one by

0:22:02.588,0:22:06.570
three will give us 15X an 15X[br]number. The equations one.

0:22:08.240,0:22:14.923
2. And I'll take equation one[br]and I will multiply it by 5 and

0:22:14.923,0:22:17.800
that will give me a new equation

0:22:17.800,0:22:24.080
3. So multiplying[br]it by 5:15

0:22:24.080,0:22:30.152
X plus 35[br]Y is equal

0:22:30.152,0:22:36.378
to 135. And then[br]equation two, I will multiply by

0:22:36.378,0:22:40.788
three and that will give me a[br]new equation for.

0:22:41.610,0:22:46.047
Oh, here we go. Multiplying this[br]by three 15X.

0:22:46.750,0:22:52.056
Plus six Y is[br]equal to 48.

0:22:53.520,0:22:59.870
These are now both 15X and[br]they're both plus 15X.

0:23:00.560,0:23:06.320
So if I take this equation away[br]from that equation, I'll have

0:23:06.320,0:23:13.040
15X minus 15X no X is at all.[br]I live eliminative, the X, I'll

0:23:13.040,0:23:18.800
just have the Wise left, so[br]let's do that equation 3 minus

0:23:18.800,0:23:25.405
equation 4. 15X takeaway[br]15X no axis 35 Y

0:23:25.405,0:23:30.965
takeaway six Y that gives[br]us 29 Y.

0:23:32.250,0:23:38.663
And then 135 takeaway 48?[br]And that's going to give us

0:23:38.663,0:23:45.076
A7 their 87 altogether. And[br]so why is 87 over 29,

0:23:45.076,0:23:47.408
which gives us 3?

0:23:48.600,0:23:55.410
Having got that, I need to know[br]the value of X so I can take

0:23:55.410,0:24:00.858
Y equals 3 and substituted back[br]let's say into equation one. So

0:24:00.858,0:24:08.122
I have 3X plus Seven times Y 7[br]threes are 21 is equal to 27 and

0:24:08.122,0:24:14.024
so 3X is 6 taking 21 away from[br]each side and access 2.

0:24:15.530,0:24:21.800
Check this in here 5 twos are[br]ten 2306 ten and six gives me 16

0:24:21.800,0:24:27.652
which is what I want so I know[br]that this is my answer. My

0:24:27.652,0:24:31.414
solution to this pair of[br]simultaneous equations and again

0:24:31.414,0:24:35.594
look how straightforward that[br]is. Much, much easier than the

0:24:35.594,0:24:40.192
first method that we saw. Also[br]think about using letters as

0:24:40.192,0:24:44.790
well. If we've got letters to[br]use instead of coefficients the

0:24:44.790,0:24:50.095
numbers here. So we might have a[br]X plus BY. Again, this is a much

0:24:50.095,0:24:53.445
better method to use. Again,[br]notice the setting down keeping

0:24:53.445,0:24:56.795
it compact, keeping the equal[br]signs under each other and

0:24:56.795,0:25:00.815
keeping a record of what we've[br]done. So do something comes out

0:25:00.815,0:25:04.165
wrong, we can check it, see what[br]we are doing.

0:25:05.130,0:25:09.303
Now all the examples that we've[br]looked at so far of all had

0:25:09.303,0:25:12.513
whole number coefficients. They[br]might have been, plus they might

0:25:12.513,0:25:15.723
be minus, but they've been whole[br]number, and everything that

0:25:15.723,0:25:19.896
we've looked at as being in this[br]sort of form XY number XY

0:25:19.896,0:25:23.427
number. Well, not all equations[br]come like that, so let's just

0:25:23.427,0:25:25.674
have a look at a couple of

0:25:25.674,0:25:29.260
examples that. Don't look like[br]the ones we've just done.

0:25:31.370,0:25:36.320
First of all, let's have this[br]One X equals 3 Y.

0:25:37.000,0:25:42.918
And X over 3 minus Y equals[br]34 pair of simultaneous

0:25:42.918,0:25:47.222
equations. Linear simultaneous[br]equations again 'cause they both

0:25:47.222,0:25:53.678
got just X&Y in an numbers,[br]nothing else, no X squared's now

0:25:53.678,0:25:55.292
ex wise etc.

0:25:56.030,0:26:01.985
We need to get them into a form[br]that we can use and that would

0:26:01.985,0:26:07.543
be nice to have XY number. So[br]let's do that with this One X

0:26:07.543,0:26:12.307
equals 3 Y, so will have X minus[br]three Y equals 0.

0:26:13.990,0:26:18.682
This one got a fraction in it.[br]Fractions we don't like, can't

0:26:18.682,0:26:22.592
handle fractions. Let's get rid[br]of the three by multiplying

0:26:22.592,0:26:26.893
everything in this equation by[br]three. So will do that three

0:26:26.893,0:26:29.630
times X over 3 just leaves us

0:26:29.630,0:26:37.120
with X. Three times the Y[br]minus three Y equals 3 times.

0:26:37.120,0:26:40.075
This going to be 102.

0:26:42.280,0:26:42.970
Problem.

0:26:44.920,0:26:48.670
These two bits here are exactly

0:26:48.670,0:26:52.408
the same. But these two[br]bits are different.

0:26:54.140,0:26:55.928
What's going to happen?

0:26:56.800,0:27:00.220
Well, clearly if we subtract[br]these two equations one from

0:27:00.220,0:27:03.640
the other, there won't be[br]anything left this side when

0:27:03.640,0:27:06.718
we've done the subtraction X[br]from X, no access.

0:27:07.750,0:27:11.533
Minus 3 Y takeaway minus three[br]Y know why is left, and yet

0:27:11.533,0:27:15.316
we're going to have 0 - 102[br]equals minus 102 at this side.

0:27:15.316,0:27:17.935
In other words, we're gonna[br]end up with that.

0:27:21.210,0:27:22.938
Which is a wee bit strange.

0:27:23.890,0:27:27.130
What's the problem? What's the[br]difficulty? Remember right back

0:27:27.130,0:27:30.730
at the beginning when we drew a[br]couple of graphs?

0:27:32.100,0:27:37.140
In the first case we had two[br]lines that actually crossed, but

0:27:37.140,0:27:41.760
in the second case I drew 2[br]lines that were parallel.

0:27:42.510,0:27:47.574
And that's exactly what we've[br]got here. We have got 2 lines

0:27:47.574,0:27:51.794
that are parallel because[br]they've got this same form. They

0:27:51.794,0:27:56.858
are parallel lines so they don't[br]meet. And what this is telling

0:27:56.858,0:28:01.922
us is there in fact is no[br]solution to this pair of

0:28:01.922,0:28:06.564
equations because they come from[br]2 parallel lines that do not

0:28:06.564,0:28:11.206
meet no solution. There isn't[br]one fixed point, so we would

0:28:11.206,0:28:12.472
write that down.

0:28:12.580,0:28:15.480
Simply say no solutions.

0:28:17.660,0:28:19.660
And it's important to keep[br]an eye out for that.

0:28:20.740,0:28:24.630
Check back, make sure the[br]arithmetic's correct yes, but do

0:28:24.630,0:28:26.186
remember that can happen.

0:28:27.140,0:28:33.720
Let's take just one more final[br]example, X over 5.

0:28:35.340,0:28:38.646
Minus Y over 4 equals 0.

0:28:40.200,0:28:47.890
3X plus 1/2 Y equals[br]70. Now for this one.

0:28:48.800,0:28:52.170
We've got fractions with[br]dominators five and four, and we

0:28:52.170,0:28:56.214
need to get rid of those. So we[br]need a common denominator.

0:28:57.050,0:29:01.160
With which we can multiply[br]everything in the equation and

0:29:01.160,0:29:06.914
those get rid of the five in the[br]fall. The obvious one to choose

0:29:06.914,0:29:13.079
is 20, because 20 is 5 times by[br]4. Let us write that down in

0:29:13.079,0:29:18.833
falls 20 times X over 5 - 20[br]times Y over 4 equals 0

0:29:18.833,0:29:21.710
be'cause. 20 * 0 is still 0.

0:29:23.050,0:29:29.206
Little bit of counseling 5[br]into 20 goes 4.

0:29:30.050,0:29:33.630
4 into 20 goes 5.

0:29:34.700,0:29:40.208
So we have 4X minus five[br]Y equals 0.

0:29:42.000,0:29:46.708
So that was our first equation[br]that was our second equation.

0:29:46.708,0:29:51.844
This one is now become our third[br]equation. So equation one has

0:29:51.844,0:29:53.556
gone to equation 3.

0:29:54.430,0:29:57.420
Let's look at equation two. Now[br]that we need to deal with it,

0:29:57.420,0:29:59.490
it's got a half way in it. So if

0:29:59.490,0:30:03.626
I multiply every. Anything by[br]two. This will become just why?

0:30:04.350,0:30:05.950
So we have 6X.

0:30:06.830,0:30:13.980
Plus Y equals 34 and so[br]equation two has become. Now

0:30:13.980,0:30:19.240
equation for. We want to[br]eliminate one of the variables

0:30:19.240,0:30:24.050
OK, which one well I'd have to[br]do quite a bit of multiplication

0:30:24.050,0:30:30.340
by 6:00 AM by 4. If it was, the[br]ex is that I wanted to get rid

0:30:30.340,0:30:35.150
of look, there's a minus five[br]here and one there, so to speak.

0:30:35.150,0:30:39.960
So if we multiply this one by[br]five, will get these two the

0:30:39.960,0:30:45.140
same. So let's do that 4X minus[br]five Y equals 0, and then times

0:30:45.140,0:30:46.620
in this by 5.

0:30:46.930,0:30:54.085
30X plus five Y equals and then[br]we do this by 5, five, 420, not

0:30:54.085,0:31:01.240
down and two to carry 5 threes[br]are 15 and the two is 17, so

0:31:01.240,0:31:07.918
that gives us 170 and now we can[br]just add these two together. So

0:31:07.918,0:31:13.642
equation three state as it was[br]equation for we multiplied by 5.

0:31:13.642,0:31:16.981
So that's gone to equation 5 and

0:31:16.981,0:31:21.920
now. Finally, we're going to[br]add together equations three

0:31:21.920,0:31:29.070
and five, and so we have 34[br]X equals 170 and wise have

0:31:29.070,0:31:30.170
been illuminated.

0:31:33.760,0:31:41.752
34 X is 170 and so[br]X is 170 / 34 and

0:31:41.752,0:31:44.416
that gives us 5.

0:31:45.430,0:31:50.050
We need to go back and[br]substituting to one of our two

0:31:50.050,0:31:53.930
equations. It's just[br]have a look which one?

0:31:55.110,0:31:57.954
Doesn't really matter, I[br]think. Actually choose to go

0:31:57.954,0:32:02.062
for that one. Why? because I[br]can see that five over 5 gives

0:32:02.062,0:32:05.538
me one, and that's a very[br]simple number. Might make the

0:32:05.538,0:32:06.802
arithmetic so much easier.

0:32:08.080,0:32:15.304
So we'll have X over 5 minus[br]Y. Over 4 equals 0. Take the

0:32:15.304,0:32:17.368
Five and substituted in.

0:32:21.520,0:32:27.895
5 over 5. That's just one, and[br]so I have one takeaway Y over 4

0:32:27.895,0:32:33.845
equals 0, so one must be equal[br]to Y over 4. If I multiply

0:32:33.845,0:32:38.945
everything by 4I end up with[br]four equals Y. So there's my

0:32:38.945,0:32:44.470
pair of answers X equals 5, Y[br]equals 4 and I really should

0:32:44.470,0:32:48.720
just check by looking at the[br]second equation now, remember.

0:32:49.050,0:32:54.117
2nd equation was 3X[br]plus 1/2 Y equals 17.

0:32:55.690,0:32:59.102
3X also, half Y

0:32:59.102,0:33:05.975
equals 17. So let's substitute[br]these in. X is 5, three X is.

0:33:05.975,0:33:11.785
Therefore AR15, three fives plus[br]1/2 of Y. But why is 4 so 1/2

0:33:11.785,0:33:18.010
of it is 2. That gives me 17,[br]which is what I want. Yes, this

0:33:18.010,0:33:23.946
is correct. Let's just recap for[br]a moment. Apparel simultaneous

0:33:23.946,0:33:29.066
equations. They represent two[br]straight lines in effect when we

0:33:29.066,0:33:34.698
solve them together, we are[br]looking for the point where the

0:33:34.698,0:33:36.746
two straight lines intersect.

0:33:38.870,0:33:43.017
The method of elimination is[br]much, much better to use than

0:33:43.017,0:33:45.279
the first method that we saw.

0:33:46.270,0:33:50.287
Remember also in the way that[br]we've set this one out. Keep a

0:33:50.287,0:33:52.450
record of what it is that you

0:33:52.450,0:33:56.636
do. Set you workout so that the[br]equal signs come under each

0:33:56.636,0:34:00.354
other and so that at a glance[br]you can look at what you've

0:34:00.354,0:34:01.498
done. Check your working.

0:34:02.080,0:34:06.460
Finally, remember the answer[br]that you get can always be

0:34:06.460,0:34:10.840
checked by substituting the pair[br]of values into the equations

0:34:10.840,0:34:15.658
that you began with. That means[br]strictly you should never get

0:34:15.658,0:34:19.162
one of these wrong. However,[br]mistakes do happen.