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The purpose of this video is to
look at the solution of

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elementary simultaneous

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equations. Before we do that,
let's just have a look at a

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relatively straightforward

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single equation. Equation we're
going to look at 2X minus

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Y equals 3.

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This is a linear equation. It's
a linear equation because there

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are no terms in it that are X
squared, Y squared or X times by

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Y or indeed ex cubes. The only
terms we've got a terms in X

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terms in Y and some numbers.

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So this represents a linear

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equation. We can rearrange it so
that it says why equal

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something, so let's just do
that. We can add Y to each side

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so that we get 2X.

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Equals 3 plus Yi. Did say add
why two each side and you might

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have wondered what happened
here. Well, if I've got minus Y

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and I add why to it, I end up
with no wise at all.

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Here we've got two X equals 3
plus Y, so let's take the three

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away from each side. 2X minus
three equals Y.

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So there I've got a nice
expression for why. If I take

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any value of X. Let's say I take
X equals 1, then why will be

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equal to two times by 1 - 3,
which gives us minus one. So for

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this value of XI, get that value
of Yi can take another value of

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XX equals 2, Y will equal.

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2 * 2 - 3, which is
plus one.

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Another value of XX equals
0 Y equals 2 times by 0 -

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3 will 2 * 0 is 0 and that
gives me minus three.

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So for every value of XI can
generate a value of Y.

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I can plot these as point
so I can plot this as the

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.1 - 1 and I can plot this
one on a graph as the .21

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and this one on a graph as
the point nort minus three.

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So let's just set that up.

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Pair of axes.

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Let's mark the values of X that
we've been having a look at.

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So that was X on there and why

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there? And let's put on the
values of why that we got.

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When X was zero hour value
of why was minus three?

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So that's. There.

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When X was one hour value was

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minus one. And when X was
two hour value was one.

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Those three points lie on
a straight line.

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Y equals 2X minus three, and
that's another reason for

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calling this a linear equation.
It gives us a straight line.

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OK.

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You've got that one. Y equals 2X
minus three. Supposing we take a

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second one 3X plus two Y equals
8, a second linear equation, and

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supposing we say these two.

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Are true at the same time.

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What does that mean?

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Well, we can plot this as a
straight line. Again, it's a

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linear equation, so it's going
to give us a straight line. Now

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I don't want to have to workout
lots of points for this, So what

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I'm going to do is just sketch
it in quickly on the graph. I'm

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going to say when X is 0.

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And cover up the Exterm 2 Y is
equal to 8, so why must be

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equal to four which is going
to be up there somewhere?

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And when? Why is 03 X is equal
to 8 and so X is 8 / 3

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which gives us 2 and 2/3. So
somewhere about their two 2/3

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and we know it's a straight
line, so we can get that by

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joining up there.

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This is the equation 3X plus two
Y equals 8. So what does it mean

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for these two to be true at the
same time? Well, it must mean

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it's this point here.

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Where the two lines cross. So
when we solve a pair of

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simultaneous equations, what
we're actually looking for is

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the intersection of two
straight lines.

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Of course, it could happen that
we have one line like that.

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And apparel line.

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They would never meet.

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And one of the examples that
we're going to be looking at

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later will show what happens
in terms of the arithmetic

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when we have this particular
case. But for now, let's go

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back and think about these
two. How can we handle these

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two algebraically so that we
don't have to draw graphs? We

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don't have to rely on
sketching, we can calculate

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which is so much easier in
most cases that actually

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drawing a graph.

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So let's take these two
equations.

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And we're going to look at two
methods of solution, so I'm

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going to look at method one.
Now, let's begin with the

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original equation that we had
two X minus. Y is equal to three

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and then the one that we put
with it 3X plus two Y equals 8.

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Our first method of solution,
well, one of the things to do is

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to do what we did in the very
first case with this and

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Rearrange one of these
equations. It doesn't matter

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which one, but we'll take this

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one. So that we get Y

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equals. And we know what the
result for that one is. It's Y

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equals 2X minus three.

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So that's equation one.

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That's equation. Two, so this
is now, let's call it equation 3

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and we got it by rearranging.

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1.

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What we're going to do with this
is if these two have to be true

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at the same time, then this
relationship must be true in

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this equation, so we can
substitute it in, so let's.

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Substitute 3

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until two so we
have 3X plus.

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Two times Y. But why is
2X minus three that's equal to

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8? And you can see that what
we've done is we've reduced.

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This. To this equation giving us
a single equation in one

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unknown, which is a simple
linear equation and we can solve

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it. Multiply out the brackets 3X
plus 4X minus 6 equals 8.

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Gather the excess together. 7X
minus 6 equals 8.

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At the six to each side, Seven X
equals 14, and so X must be 2.

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That's only given as one value.

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We need a value of Y, but up
here we've got an expression

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which says Y equals and if we
take the value of X that we've

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got and substituted in.

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Therefore, why
will be equal to 2

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* 2 - 3 gives US1?

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And so we've got a solution X
equals 2, Y equals 1.

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Are we sure it's right?

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Well, we used this equation
which came from equation one to

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generate the value of Y.

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So if we take the values of X&Y
and put them back into here,

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they should work, should give us
the right answer. So let's try

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that. X is 2.

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3 times by two is 6 plus, Y is
1 two times by one is 2 six and

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two gives us eight. Yes, this
works. This is a solution of

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that equation and of that one.
So this is our answer to the

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pair of simultaneous equations.

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Let's have a look at another one
using this particular method.

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The example we're going to use
is going to be said.

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Open X. +2
Y equals 47 and

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five X minus four
Y equals 1.

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Now. We need to make a choice.
We need to choose one of these

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two equations. And Rearrange it
so that it says Y equals or if

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we want X equals.

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The choice is entirely ours and
we have to make the choice based

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upon what we feel will be the
simplest and looking at a pair

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of equations like this often
difficult to know which is the

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simplest. Well, let's pick at
random. Let's choose this one

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and let's Rearrange Equation
too. So we'll start by getting X

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equals this time. So we say 5X
is equal to 1 and I'm going to

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add 4 Y to each side plus 4Y.

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Now I'm going to divide
throughout by the five so that I

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have. X on its own. Now I've got
to divide everything by 5.

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Everything so I had to put
that line there to show that

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I'm dividing the one and the
four Y. So this is a fraction.

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I'm sure you can tell this is
not going to be as easy as the

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previous question was. In
fact, it's going to be quite

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difficult because I have to
take this now and because it

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came from equation too.

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I'm going to have to take it and
substitute it back into equation

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one, and this isn't looking very
pretty, so let's give it a try

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sub. 3.

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Until one. So I
have 7X but X is this

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lump of algebra here 1 +
4 Y all over 5.

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+2 Y equals

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47. I can see this is
becoming quite horrific.

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Multiply throughout by 5 why?
Because we're dividing by 5. We

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want to get rid of the fraction.
The way to do that is to

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multiply everything by 5 and it
has to be everything. So if we

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multiply that by 5 because we're
dividing by 5, it's as though we

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actually do nothing to the 1 + 4
Y. That leaves a 7 * 1 + 4 Y.

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We need five times that that's
ten Y and we have to have five

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times that remember, an
equation is a balance. What

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you do to one side of the
balance you have to do to the

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other. If you don't, it's
unbalanced. So we're

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multiplying everything by 5.
So 5 * 47 five 735, five falls

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of 22135 altogether.

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Now we need to multiply out the
brackets 7 + 28 Y plus 10

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Y equals 235. So we take this

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equation. Write it down again so
that we can see it clearly.

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Now we can gather these two
together gives us 38Y.

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And we can take Seven away
from each side, which will

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give us 228.

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Exactly big numbers coming in
here 228 / 38 'cause we're

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looking for the number which
when we multiplied by 38 will

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give us 228 and that's going to

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be 6. So we've established Y is
equal to 6.

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Having done that, we can take it
and we can substitute it back

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into the equation that we first
had for X. So remember that for

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that we had X was equal to.

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And what we had for that was 1
+ 4 Y all over five. We

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substitute in the six, so we
have 1 + 24 or over 5 and

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quickly we can see that's 25 /
5. So we have X equals 5. So

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again we've got our pair of
values. Our answer to the pair

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of simultaneous equations. We
haven't checked it though.

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Now remember that this came from
the second equation, so really

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to check it we've got to go back
to the very first equation that

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we had written down that one. If
you remember was Seven X +2, Y

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is equal to 47.

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So let's just check 7 *
5. That gives us 35 +

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2 * 6.

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That gives us 12, so we 35 + 12
equals 47. And yes, that is what

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we wanted, so we now know that
this is correct, but I just stop

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and think about it.

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We got all those fractions to
work with. We got this lump of

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algebra to carry around with us.
Is there not an easier way of

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doing these? Yes there is. It's
useful to have seen the method

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that we have got simply because
we will need it again when we

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look at the second video of
simultaneous equations, but.

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That is a simple way of handling
these, so let's go on now and

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have a look at method 2.

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Now this method is sometimes
called elimination and we can

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see why it gets that name and
this is the method that you

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really do need to practice and
become accustomed to. So let's

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start with the same equations
that we had last time.

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And see.

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How it works and how much easier
it actually is?

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OK method of elimination. What
do we do?

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What we do is we seek
to make the

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coefficients in front
of the wise.

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Or in front of the axes.

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The same.

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Once we've gotten the same,
then we can either add the

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two equations together or
subtract them according to

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the signs that are there.

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By doing that, we will get rid
of that particular unknown, the

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one that we chose.

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To make the coefficients
numerically the same.

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So. This one what would we do?

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Well, if we look at this and

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this. Here we have two Y and
here we have minus four Y.

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So if I were to double that, I'd
have four. Why there? And it's

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plus four Y Ana minus four Y
there, and that seems are pretty

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good thing to do, because then
they're both for Y. One of them

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is plus and one of them is
minus. And if I add them

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together they will disappear. So
let me just number the equations

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one and two.

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And then I can keep a record of
what I'm doing. So I'm going to

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multiply the first equation by
two and that's going to lead

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Maine to a new equation 3.

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So let's do that.

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2 * 7 X is 14X
plus two times, that is 4

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Y equals 2 times that, and
2 * 47 is 94.

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Now equation two. I'm leaving as
it is not going to touch it.

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Now I've got two equations.

231
00:18:34,110 --> 00:18:41,194
This is plus four Y and this
is minus four Y. So if I

232
00:18:41,194 --> 00:18:47,266
add the two equations together,
what happens? I get 14X plus 5X.

233
00:18:47,266 --> 00:18:48,784
That's 19 X.

234
00:18:50,880 --> 00:18:56,775
No whys. 'cause I've plus four Y
add it to minus four Y know wise

235
00:18:56,775 --> 00:18:57,954
at all equals.

236
00:18:58,820 --> 00:18:59,850
95

237
00:19:01,010 --> 00:19:06,386
and so X is 95 over 19, which
gives me 5 which if you

238
00:19:06,386 --> 00:19:10,226
remember, is the answer we have
to the last question.

239
00:19:11,820 --> 00:19:15,285
Now we need to take this and
substitute it back. Doesn't

240
00:19:15,285 --> 00:19:18,435
matter which equation we choose
to substitute it back into.

241
00:19:18,435 --> 00:19:19,695
Let's take this one.

242
00:19:20,350 --> 00:19:27,562
X is 5, so five times
by 5 - 4 Y equals

243
00:19:27,562 --> 00:19:28,163
1.

244
00:19:29,200 --> 00:19:35,090
And so we have 25 -
4 Y equals 1.

245
00:19:37,200 --> 00:19:39,930
Take the four way over
to that side by adding

246
00:19:39,930 --> 00:19:41,295
four Y to each side.

247
00:19:43,260 --> 00:19:49,116
So that will give us 25 is equal
to four Y plus one. Take the one

248
00:19:49,116 --> 00:19:54,972
away from its side, 24 is 4 Y
and so why is equal to 6 and

249
00:19:54,972 --> 00:19:58,998
we've got exactly the same
answer as we had before. And

250
00:19:58,998 --> 00:20:00,096
let's just look.

251
00:20:01,400 --> 00:20:05,426
How much simpler that is? How
much quicker that answer came

252
00:20:05,426 --> 00:20:09,818
out. One thing to notice. Well,
two things in actual fact. First

253
00:20:09,818 --> 00:20:14,210
of all, I try to keep the equal
signs underneath each other.

254
00:20:15,060 --> 00:20:19,246
This is not only makes it look
neat, it enables you to see what

255
00:20:19,246 --> 00:20:20,442
it is you're doing.

256
00:20:21,540 --> 00:20:25,082
Keep the equations together so
the setting out of this work

257
00:20:25,082 --> 00:20:27,336
actually helps you to be able to

258
00:20:27,336 --> 00:20:32,783
check it. Second thing to notice
is down this side. I've kept a

259
00:20:32,783 --> 00:20:36,886
record of exactly what I've done
multiplying the equation by two,

260
00:20:36,886 --> 00:20:39,870
adding the two equations
together. That's very helpful

261
00:20:39,870 --> 00:20:45,092
when you want to check your
work. What did I do? How did I

262
00:20:45,092 --> 00:20:48,822
actually work this out? By
having this record down the

263
00:20:48,822 --> 00:20:53,671
side, you don't have to work it
out again. You can see exactly

264
00:20:53,671 --> 00:20:58,520
what it is that you did. Now
let's take a third example and

265
00:20:58,520 --> 00:21:03,218
again. Will solve it by means of
the method of elimination. Just

266
00:21:03,218 --> 00:21:08,650
so we've got a second example of
that method to look at three X

267
00:21:08,650 --> 00:21:10,978
+7 Y is 27 and 5X.

268
00:21:12,620 --> 00:21:14,968
+2 Y is 60.

269
00:21:16,430 --> 00:21:20,234
OK, we've got a choice to make.
We can make either the

270
00:21:20,234 --> 00:21:23,721
coefficients in front of the axe
numerically the same, or the

271
00:21:23,721 --> 00:21:27,525
coefficients in front of the
wise. Well, in order to do that,

272
00:21:27,525 --> 00:21:31,012
I'd have to multiply the Y.
Certainly have to multiply this

273
00:21:31,012 --> 00:21:35,450
equation by two to give me 14
there and this one by 7:00 to

274
00:21:35,450 --> 00:21:36,718
give me 14 there.

275
00:21:38,310 --> 00:21:39,934
How do I make that choice? Well?

276
00:21:40,990 --> 00:21:45,742
Fairly clearly 2 times by 7 is
14, so 1 by 1, one by the other.

277
00:21:46,660 --> 00:21:49,918
But I don't really like
multiplying by 7 difficult

278
00:21:49,918 --> 00:21:54,262
number. I prefer to multiply by
three and five, so my choices

279
00:21:54,262 --> 00:21:58,244
actually governed by how well I
think I can handle the

280
00:21:58,244 --> 00:22:02,588
arithmetic. So let's multiply
this one by 5 and this one by

281
00:22:02,588 --> 00:22:06,570
three will give us 15X an 15X
number. The equations one.

282
00:22:08,240 --> 00:22:14,923
2. And I'll take equation one
and I will multiply it by 5 and

283
00:22:14,923 --> 00:22:17,800
that will give me a new equation

284
00:22:17,800 --> 00:22:24,080
3. So multiplying
it by 5:15

285
00:22:24,080 --> 00:22:30,152
X plus 35
Y is equal

286
00:22:30,152 --> 00:22:36,378
to 135. And then
equation two, I will multiply by

287
00:22:36,378 --> 00:22:40,788
three and that will give me a
new equation for.

288
00:22:41,610 --> 00:22:46,047
Oh, here we go. Multiplying this
by three 15X.

289
00:22:46,750 --> 00:22:52,056
Plus six Y is
equal to 48.

290
00:22:53,520 --> 00:22:59,870
These are now both 15X and
they're both plus 15X.

291
00:23:00,560 --> 00:23:06,320
So if I take this equation away
from that equation, I'll have

292
00:23:06,320 --> 00:23:13,040
15X minus 15X no X is at all.
I live eliminative, the X, I'll

293
00:23:13,040 --> 00:23:18,800
just have the Wise left, so
let's do that equation 3 minus

294
00:23:18,800 --> 00:23:25,405
equation 4. 15X takeaway
15X no axis 35 Y

295
00:23:25,405 --> 00:23:30,965
takeaway six Y that gives
us 29 Y.

296
00:23:32,250 --> 00:23:38,663
And then 135 takeaway 48?
And that's going to give us

297
00:23:38,663 --> 00:23:45,076
A7 their 87 altogether. And
so why is 87 over 29,

298
00:23:45,076 --> 00:23:47,408
which gives us 3?

299
00:23:48,600 --> 00:23:55,410
Having got that, I need to know
the value of X so I can take

300
00:23:55,410 --> 00:24:00,858
Y equals 3 and substituted back
let's say into equation one. So

301
00:24:00,858 --> 00:24:08,122
I have 3X plus Seven times Y 7
threes are 21 is equal to 27 and

302
00:24:08,122 --> 00:24:14,024
so 3X is 6 taking 21 away from
each side and access 2.

303
00:24:15,530 --> 00:24:21,800
Check this in here 5 twos are
ten 2306 ten and six gives me 16

304
00:24:21,800 --> 00:24:27,652
which is what I want so I know
that this is my answer. My

305
00:24:27,652 --> 00:24:31,414
solution to this pair of
simultaneous equations and again

306
00:24:31,414 --> 00:24:35,594
look how straightforward that
is. Much, much easier than the

307
00:24:35,594 --> 00:24:40,192
first method that we saw. Also
think about using letters as

308
00:24:40,192 --> 00:24:44,790
well. If we've got letters to
use instead of coefficients the

309
00:24:44,790 --> 00:24:50,095
numbers here. So we might have a
X plus BY. Again, this is a much

310
00:24:50,095 --> 00:24:53,445
better method to use. Again,
notice the setting down keeping

311
00:24:53,445 --> 00:24:56,795
it compact, keeping the equal
signs under each other and

312
00:24:56,795 --> 00:25:00,815
keeping a record of what we've
done. So do something comes out

313
00:25:00,815 --> 00:25:04,165
wrong, we can check it, see what
we are doing.

314
00:25:05,130 --> 00:25:09,303
Now all the examples that we've
looked at so far of all had

315
00:25:09,303 --> 00:25:12,513
whole number coefficients. They
might have been, plus they might

316
00:25:12,513 --> 00:25:15,723
be minus, but they've been whole
number, and everything that

317
00:25:15,723 --> 00:25:19,896
we've looked at as being in this
sort of form XY number XY

318
00:25:19,896 --> 00:25:23,427
number. Well, not all equations
come like that, so let's just

319
00:25:23,427 --> 00:25:25,674
have a look at a couple of

320
00:25:25,674 --> 00:25:29,260
examples that. Don't look like
the ones we've just done.

321
00:25:31,370 --> 00:25:36,320
First of all, let's have this
One X equals 3 Y.

322
00:25:37,000 --> 00:25:42,918
And X over 3 minus Y equals
34 pair of simultaneous

323
00:25:42,918 --> 00:25:47,222
equations. Linear simultaneous
equations again 'cause they both

324
00:25:47,222 --> 00:25:53,678
got just X&Y in an numbers,
nothing else, no X squared's now

325
00:25:53,678 --> 00:25:55,292
ex wise etc.

326
00:25:56,030 --> 00:26:01,985
We need to get them into a form
that we can use and that would

327
00:26:01,985 --> 00:26:07,543
be nice to have XY number. So
let's do that with this One X

328
00:26:07,543 --> 00:26:12,307
equals 3 Y, so will have X minus
three Y equals 0.

329
00:26:13,990 --> 00:26:18,682
This one got a fraction in it.
Fractions we don't like, can't

330
00:26:18,682 --> 00:26:22,592
handle fractions. Let's get rid
of the three by multiplying

331
00:26:22,592 --> 00:26:26,893
everything in this equation by
three. So will do that three

332
00:26:26,893 --> 00:26:29,630
times X over 3 just leaves us

333
00:26:29,630 --> 00:26:37,120
with X. Three times the Y
minus three Y equals 3 times.

334
00:26:37,120 --> 00:26:40,075
This going to be 102.

335
00:26:42,280 --> 00:26:42,970
Problem.

336
00:26:44,920 --> 00:26:48,670
These two bits here are exactly

337
00:26:48,670 --> 00:26:52,408
the same. But these two
bits are different.

338
00:26:54,140 --> 00:26:55,928
What's going to happen?

339
00:26:56,800 --> 00:27:00,220
Well, clearly if we subtract
these two equations one from

340
00:27:00,220 --> 00:27:03,640
the other, there won't be
anything left this side when

341
00:27:03,640 --> 00:27:06,718
we've done the subtraction X
from X, no access.

342
00:27:07,750 --> 00:27:11,533
Minus 3 Y takeaway minus three
Y know why is left, and yet

343
00:27:11,533 --> 00:27:15,316
we're going to have 0 - 102
equals minus 102 at this side.

344
00:27:15,316 --> 00:27:17,935
In other words, we're gonna
end up with that.

345
00:27:21,210 --> 00:27:22,938
Which is a wee bit strange.

346
00:27:23,890 --> 00:27:27,130
What's the problem? What's the
difficulty? Remember right back

347
00:27:27,130 --> 00:27:30,730
at the beginning when we drew a
couple of graphs?

348
00:27:32,100 --> 00:27:37,140
In the first case we had two
lines that actually crossed, but

349
00:27:37,140 --> 00:27:41,760
in the second case I drew 2
lines that were parallel.

350
00:27:42,510 --> 00:27:47,574
And that's exactly what we've
got here. We have got 2 lines

351
00:27:47,574 --> 00:27:51,794
that are parallel because
they've got this same form. They

352
00:27:51,794 --> 00:27:56,858
are parallel lines so they don't
meet. And what this is telling

353
00:27:56,858 --> 00:28:01,922
us is there in fact is no
solution to this pair of

354
00:28:01,922 --> 00:28:06,564
equations because they come from
2 parallel lines that do not

355
00:28:06,564 --> 00:28:11,206
meet no solution. There isn't
one fixed point, so we would

356
00:28:11,206 --> 00:28:12,472
write that down.

357
00:28:12,580 --> 00:28:15,480
Simply say no solutions.

358
00:28:17,660 --> 00:28:19,660
And it's important to keep
an eye out for that.

359
00:28:20,740 --> 00:28:24,630
Check back, make sure the
arithmetic's correct yes, but do

360
00:28:24,630 --> 00:28:26,186
remember that can happen.

361
00:28:27,140 --> 00:28:33,720
Let's take just one more final
example, X over 5.

362
00:28:35,340 --> 00:28:38,646
Minus Y over 4 equals 0.

363
00:28:40,200 --> 00:28:47,890
3X plus 1/2 Y equals
70. Now for this one.

364
00:28:48,800 --> 00:28:52,170
We've got fractions with
dominators five and four, and we

365
00:28:52,170 --> 00:28:56,214
need to get rid of those. So we
need a common denominator.

366
00:28:57,050 --> 00:29:01,160
With which we can multiply
everything in the equation and

367
00:29:01,160 --> 00:29:06,914
those get rid of the five in the
fall. The obvious one to choose

368
00:29:06,914 --> 00:29:13,079
is 20, because 20 is 5 times by
4. Let us write that down in

369
00:29:13,079 --> 00:29:18,833
falls 20 times X over 5 - 20
times Y over 4 equals 0

370
00:29:18,833 --> 00:29:21,710
be'cause. 20 * 0 is still 0.

371
00:29:23,050 --> 00:29:29,206
Little bit of counseling 5
into 20 goes 4.

372
00:29:30,050 --> 00:29:33,630
4 into 20 goes 5.

373
00:29:34,700 --> 00:29:40,208
So we have 4X minus five
Y equals 0.

374
00:29:42,000 --> 00:29:46,708
So that was our first equation
that was our second equation.

375
00:29:46,708 --> 00:29:51,844
This one is now become our third
equation. So equation one has

376
00:29:51,844 --> 00:29:53,556
gone to equation 3.

377
00:29:54,430 --> 00:29:57,420
Let's look at equation two. Now
that we need to deal with it,

378
00:29:57,420 --> 00:29:59,490
it's got a half way in it. So if

379
00:29:59,490 --> 00:30:03,626
I multiply every. Anything by
two. This will become just why?

380
00:30:04,350 --> 00:30:05,950
So we have 6X.

381
00:30:06,830 --> 00:30:13,980
Plus Y equals 34 and so
equation two has become. Now

382
00:30:13,980 --> 00:30:19,240
equation for. We want to
eliminate one of the variables

383
00:30:19,240 --> 00:30:24,050
OK, which one well I'd have to
do quite a bit of multiplication

384
00:30:24,050 --> 00:30:30,340
by 6:00 AM by 4. If it was, the
ex is that I wanted to get rid

385
00:30:30,340 --> 00:30:35,150
of look, there's a minus five
here and one there, so to speak.

386
00:30:35,150 --> 00:30:39,960
So if we multiply this one by
five, will get these two the

387
00:30:39,960 --> 00:30:45,140
same. So let's do that 4X minus
five Y equals 0, and then times

388
00:30:45,140 --> 00:30:46,620
in this by 5.

389
00:30:46,930 --> 00:30:54,085
30X plus five Y equals and then
we do this by 5, five, 420, not

390
00:30:54,085 --> 00:31:01,240
down and two to carry 5 threes
are 15 and the two is 17, so

391
00:31:01,240 --> 00:31:07,918
that gives us 170 and now we can
just add these two together. So

392
00:31:07,918 --> 00:31:13,642
equation three state as it was
equation for we multiplied by 5.

393
00:31:13,642 --> 00:31:16,981
So that's gone to equation 5 and

394
00:31:16,981 --> 00:31:21,920
now. Finally, we're going to
add together equations three

395
00:31:21,920 --> 00:31:29,070
and five, and so we have 34
X equals 170 and wise have

396
00:31:29,070 --> 00:31:30,170
been illuminated.

397
00:31:33,760 --> 00:31:41,752
34 X is 170 and so
X is 170 / 34 and

398
00:31:41,752 --> 00:31:44,416
that gives us 5.

399
00:31:45,430 --> 00:31:50,050
We need to go back and
substituting to one of our two

400
00:31:50,050 --> 00:31:53,930
equations. It's just
have a look which one?

401
00:31:55,110 --> 00:31:57,954
Doesn't really matter, I
think. Actually choose to go

402
00:31:57,954 --> 00:32:02,062
for that one. Why? because I
can see that five over 5 gives

403
00:32:02,062 --> 00:32:05,538
me one, and that's a very
simple number. Might make the

404
00:32:05,538 --> 00:32:06,802
arithmetic so much easier.

405
00:32:08,080 --> 00:32:15,304
So we'll have X over 5 minus
Y. Over 4 equals 0. Take the

406
00:32:15,304 --> 00:32:17,368
Five and substituted in.

407
00:32:21,520 --> 00:32:27,895
5 over 5. That's just one, and
so I have one takeaway Y over 4

408
00:32:27,895 --> 00:32:33,845
equals 0, so one must be equal
to Y over 4. If I multiply

409
00:32:33,845 --> 00:32:38,945
everything by 4I end up with
four equals Y. So there's my

410
00:32:38,945 --> 00:32:44,470
pair of answers X equals 5, Y
equals 4 and I really should

411
00:32:44,470 --> 00:32:48,720
just check by looking at the
second equation now, remember.

412
00:32:49,050 --> 00:32:54,117
2nd equation was 3X
plus 1/2 Y equals 17.

413
00:32:55,690 --> 00:32:59,102
3X also, half Y

414
00:32:59,102 --> 00:33:05,975
equals 17. So let's substitute
these in. X is 5, three X is.

415
00:33:05,975 --> 00:33:11,785
Therefore AR15, three fives plus
1/2 of Y. But why is 4 so 1/2

416
00:33:11,785 --> 00:33:18,010
of it is 2. That gives me 17,
which is what I want. Yes, this

417
00:33:18,010 --> 00:33:23,946
is correct. Let's just recap for
a moment. Apparel simultaneous

418
00:33:23,946 --> 00:33:29,066
equations. They represent two
straight lines in effect when we

419
00:33:29,066 --> 00:33:34,698
solve them together, we are
looking for the point where the

420
00:33:34,698 --> 00:33:36,746
two straight lines intersect.

421
00:33:38,870 --> 00:33:43,017
The method of elimination is
much, much better to use than

422
00:33:43,017 --> 00:33:45,279
the first method that we saw.

423
00:33:46,270 --> 00:33:50,287
Remember also in the way that
we've set this one out. Keep a

424
00:33:50,287 --> 00:33:52,450
record of what it is that you

425
00:33:52,450 --> 00:33:56,636
do. Set you workout so that the
equal signs come under each

426
00:33:56,636 --> 00:34:00,354
other and so that at a glance
you can look at what you've

427
00:34:00,354 --> 00:34:01,498
done. Check your working.

428
00:34:02,080 --> 00:34:06,460
Finally, remember the answer
that you get can always be

429
00:34:06,460 --> 00:34:10,840
checked by substituting the pair
of values into the equations

430
00:34:10,840 --> 00:34:15,658
that you began with. That means
strictly you should never get

431
00:34:15,658 --> 00:34:19,162
one of these wrong. However,
mistakes do happen.