In this session we're going to be having a look at simple equations
in one variable and the equations will be linear.
That means that there are no x-squared terms
and no x-cube terms, just x's and numbers. So let's have a
look at the first one. 3x plus 15 equals x plus 25.
An important thing to remember about any equation is this equal
sign represents a balance. What that equal sign says is that
what's on the left hand side is exactly the same as what's on
the right hand side.
If we do anything to one side of the equation, we have to do it
to the other side. If we don't, the balance is disturbed.
If we can keep that in mind,
then whatever operations we perform on either side of the
equation, so long as it's done in exactly the same way on other side,
we should be alright.
Our first step in solving any equation is to attempt to gather
all the x terms together and all these free floating numbers together.
To begin with we've got 3x on the left and and x on the right.
If we take an x away from both sides, we take one x off the left,
and one x off the right. That will give us 2x plus 15 equals 25.
We need to get the numbers together. These free numbers.
We can see that if we take 15 away from the left, and 15 away from the right
then we will have no numbers remaining on the left, just a 2x
and taking 15 away on the right, that gives us 10.
So, x must be equal to 5.
That was relatively straightforward.
Lots of plus signs, no minus signs, which we know can be complicated,
also no brackets. So let's introduce both of those.
We will take 2x plus 3 is equal to
6 minus bracket 2x minus three bracket.
Now this right hand side needs a little bit of dealing with.
It needs getting into shape so we have to remove
this bracket. 2x plus 3 equals 6,
now we take away 2x, thats minus 2x and then we're taking away minus 3.
When we take away minus 3, that makes it plus 3.
Before we go any further, we need to tidy this right hand
side up a little bit more 2x plus 3 is equal to 9 minus 2x
and now we're in the same position at this line as we were
when we started there. So we need to get the xs together,
which we can best do by adding 2x to each side.
On the right, minus 2x plus 2x. This side gives us no x.
And on the left it's going to give us 4X.
4X plus 3 is equal to 9 because minus 2X plus 2X gives no x.
Now I can take 3 away from each side, 4X equals 6
and so x is 6 divided by 4 and we want to write that in its
lowest terms which is 3 over 2. Perfectly acceptable answer.
Or one and a half. So either of these are acceptable answers.
This one (6 over 4) isn't because it's not in its lowest terms, it must be reduced
to its lowest terms. So either of those two are OK as answers.
With all of these equations, we should strictly check back by
taking our answer and putting it into the first line of the
equation and seeing if we get the right answer.
So with this five, 3 fives are 15, and 15 is 30. 5 plus 25 is also 30
The balance that we talked about at the beginning is maintained.
If you take this number, 3 over 2, or one and a half.
And substitute it back into here we should find again that
it gives us the right answer, that both sides give the same value.
They balance. Solet's just do that.
Let us take one and a half. So 2 times one and a half is 3,
plus three is 6, so we've got six on the left side.
Here we have 6, takeaway, now let's deal with this bracket,
just this bracket on its own, nothing else. 2x takeaway, three, that's nothing,
so we are just left with 6. We calculated this side to be 6.
Six equals six, so again, we've got that
balance, so we know that we've got the right answer.
Let's carry on and have a look at one or two questions with
more brackets and this time numbers outside those
brackets as well. So in a sense, what we're doing is we are
increasing the complexity of the equation, but the simple
principles that we've got so far are going to help us out because
no matter how complicated it gets and this does look
complicated, the same ideas work all the time.
We begin by multiplying out the brackets and taking care,
in particular, with any minus signs that come up so
8 times by x is 8x and eight times by minus three is minus 24.
We've multiplied everything inside that bracket by what's
outside, so we've 8 times by x and we've 8 times by minus 3
Now we remove this bracket were taking away 6,
that's minus six and we're taking away minus 2x,
a minus minus gives us a plus. So that's plus two X
equals, two times x here, is 2x, 2 times by 2 is 4.
and now we have to multiply by minus five, so we've got
minus five times by 5. That's minus 25. And minus five times by minus x.
So that's plus 5x.
Each side needs tidying up. We need to look at this side and
gather x terms and numbers together. so with 8x plus 2x that's 10x.
Take away 24, takeaway six, that's taking away 30 altogether.
2x and 5x gives us 7x.
Add on 4 takeaway 25. Now that's the equivalent of taking
away 21. Let's get the xs together. We can take 7x away from each side
so we would have 3x minus 30 there equals just
minus 21 because we've taken that 7x away.
Add the 30 to each side, because minus 30 + 30 gives us
nothing. And add it on over here. So we get three x equals,
30 added to minus 21, just the same as 30 takeaway 21,
the answer is 9.
And three times by something to give us 9 means the
something, the x, must be 3. Again we ought to be able
to take that 3, put it back into this line and see that
in fact we've got the correct answer. So let's just try.
3 minus 3 is zero. 8 times by 0 is still 0,
so we can forget about that term.
2 times by 3three is 6, so six takeaway six is nothing, so
we've actually got nothing there, zero, and nothing there, zero.
so there's nothing on the left side of the equation.
The right hand side should come out to 0 as well. Let's see that it does.
3 plus 2 is 5 and 2 fives are 10. 5 takeaway 3 is 2,
and 5 twos are 10. So we've got 10 takeaway 10,
this gives us nothing again, so we've got nothing on the right.
The equation balances with this value,
so this is our one and only solution.
We take a final example of this kind with
x plus 1 times by 2x plus 1.
Equals x plus 3 times by 2x plus 3
- 14.
Now, I did say that these were linear equations that there
would be no x-squared terms in them.
But when we start to multiply out these
brackets, we are going to get some x-squared terms.
Let's have a look, to show you what actually does happen.
We do x times by two x. That's two x-squared.
x times by one, that is plus x.
One times two x, that is plus 2x.
1 times by 1
+ 1. Equals...
x times by two x, that's two x-squared.
x times by three is 3x..
3 times by 2x is 6x.
3 times by 3 is 9 and finally takeaway 14.
Now we need to tidy up both sides here.
Two x-squared plus 3x plus one equals
2 x-squared plus 9x, (taking these two terms together)
Plus 9, minus 14.
And this needs a little bit more tidying, but before we do that,
let's just have a look at what we've got.
We've got 2 x-squared there, and 2 x-squared there. They are both
positive, so I can take two x-sqaured away from both sides.
That means that the two x-squared vanishes from both
sides, so let's do that. Take two x-squared from each side,
and that leaves us 3x plus one, is equal to 9x, and now we can
do this bit 9 takeaway 14,
Well, that's going to be takeaway five. I still have the
five to subtract. Now we're back to where we used to be getting
the xs together, getting the numbers together. We can
subtract 3x from both sides, so that gives me one equals 6x minus 5
We can add the five to both sides so that
6 equals 6X, and so 1 is equal to x.
And again, we can check that this works. We can substitute it back in.
1 plus 1 is 2, two ones are two, and one is 3.
So effectively we've got 3 times by two at this side, which gives 6.
Here 1 + 3 is 4 two times by one is 2, + 3 is 5
so we've got four times by 5. So altogether there there's 20.
Takeaway 14 is again 6, so again, this equation is balanced
exactly when we take x to be equal to 1.
Now, those equations that we've just looked at have really been
about whole numbers. The coefficients have been whole
numbers. Everything's been in terms of integers. What happens
when we start to get some fractions in there? Some
rational numbers? How do we deal with that? So again, we're going
to add in more complexity, but again, the rules are the same.
What we do to one side of the equation we must do to the other
side in order to preserve that
particular balance. So let's introduce some fractions
along with some brackets.
4 bracket X plus 2, all over 5 is equal to
7 plus 5x over 13.
We've got some fractions here.
Numbers in the denominator 5 and 30. We want rid of those we want
to be able to work with whole numbers with integers, so we
have to find a way of getting rid of them. That means we
have to multiply everything because what we do to one side,
we must do to the other.
The common denominator for five and 13 is 65, that's five times by 13.
So let's do that. Let's multiply everything by 65 and
I'll write it down in full so that we can see it happening.
we have 65 times 4, brackets x+2 over 5. Then we have 65 times by 7.
Remember, I said we have to multiply everything, so it's not
just these fraction bits, it's any spare numbers that there are
around as well.
Plus 65 times by 5x over 30.
Now let's look at each term and make it simpler. Tidy it up.
For a start, five will divide into 65, so 5 into five goes one
and five into 65 goes 13 times.
And then four times by 13? Well, that's 52, so we have
52 times by x +2 equals.
That's in a nice familiar form where used to that sort of
former. We've arrived at it by choosing to multiply everything
by this common denominator. Let's tidy this side up.
7 times by 65. Well, that's pretty tall order. Let's try it.
Seven 5s are 35. Seven 6s are 42 and three is 45.
Here with 65 times 5 x over 13. 13 goes into 13 once,
and 13 goes into 65 five times. So we five times by 5x is 25x.
You may say 'what happened to these ones?'. Well if I divide by
one it stays unchanged so I don't have to write them down.
And now we left with an equation that were used to
handling. We've met these kind before, so let's multiply out
the brackets, get the xs together and solve the equation.
So we multiply out the brackets 52 times 2 is 104
equals 455 plus 25x. Take the 25x away from each side,
gives me 27x there.
And no x is there. Take the 104 away
from each side, which gives me 351.
And now lots of big numbers, really, that shouldn't be a
problem to us. 351 divided by 27 will certainly go once
(there's one 27 in the 35 and eight over and 27s into 8
go three times, so answer is x equals 13 and we
should go back and check it and make sure that it's right.
So let's do that.
13 + 2? Well, that's 15
15 divided 5 is 3 and now times by 4, so that's 12.
Hang on to that number 12 at that side.
5 times 13 divided by 13. So the answer is just five
plus the 7 is again 12. The same as this side.
So the answer is correct. It balances.
Let's practice that one again and have a look at another example.
We take X plus 5 over 6
minus x plus one over 9
is equal to x plus three over 4.
We haven't got any brackets.
Does that make any difference? The thing you have
to remember is that this line.
Not only acts as a division sign, but it acts as a bracket.
It means that all of x plus 5 is divided by 6.
So it might be as well if we kept that in mind and put
brackets around these terms so that we're clear that
we've written down that these are to be kept together and are
all divided by 6. These two are to be kept together and all
divided by 9. And similarly here they are all divided by 4.
Next step we need a common denominator. We need a number
into which all of these will divide exactly. Now we could
multiply them altogether and we'd be certain.
But the arithmetic would be horrendous. 6 times 9 times 4 is very big.
Can we find a smaller number into which six, nine, and
four will all divide?
Well, a candidate for that is 36. 36 will divide by 6l
36 will divide by 9 and 36 will divide by 4, so lets multiply
throughout by that number 36. We have 36, because we've put the
brackets in we are quite clear that we're multiplying
everything over that 6 by the 36. Minus 36 times x plus one.
All over 9 equals 36 times by x, plus three all over 4, so we
made it quite clear by using the brackets what this 36 is
multiplying . 6 into six goes once and six into 36 goes 6 times.
9 into nine goes once and 9 into 36 goes 4.
4 into 4 goes once and four into 36 goes 9.
So now I have this bracket to multiply by 6 this bracket to
multiply by 4 and this bracket to multiply by 9. I don't have
to worry about the ones because I'm dividing by them so they
leave everything unchanged. So let's multiply out six times by
6X plus 30 (6 times by 5). This is a minus four I'm
multiplied by, so I need to be a
bit careful. Minus four times x is minus 4x,
Minus 4 times 1 is minus 4
Equals 9X and 9 threes
are 27. So now we need to simplify this side
6x takeaway 4x. That's just two X30 takeaway, four is 26, and
that's equal to 9X plus 27.
Let me take 2X away from each side, so I have 26 equals 7X
plus 27 and now I'll take the Seven away from each side and
I'll have minus one is equal to 7X and so now I need to divide
both sides by 7 and so I get minus 7th for my answer. Don't
worry that this is a fraction, sometimes they workout like
that. Don't worry, that is the negative number. Sometimes they
workout like that. Let's have a look at another one 'cause this
is a process that you're going to have to be able to do quite
complicated questions. So we'll take 4 - 5 X.
All over 6.
Minus 1 - 2 X all over
3. Equals 13 over 42.
What are we going to do? First of all, let's remind ourselves
that this line.
Not only means divide.
Divide 4 - 5 X by 6 but it means divide all of 4 - 5 X by 6. So
let's put it in a bracket to remind ourselves and let's do
the same there.
Now we need a common denominator, six and three and
40. Two, well, six goes into 42 and three goes into 42 as well.
So let's choose 42 as our denominator, an multiply
everything by 42. So will have 42 * 4 - 5 X or
over 6 - 42 * 1 - 2 X all over 3.
Equals 42 * 13 over 42.
Six goes into six once and six goes into 42 Seven times.
Three goes into three once and free goes into 4214 times.
42 goes into itself once and
again once. So now I need to multiply out these brackets.
And simplify this side.
So 7 times by 4 gives us 28 Seven times. My minus five
gives us minus 35 X. Now here we have a minus sign and the
14, so it's minus 14 times by 1 - 14 and then it's minus
14 times by minus 2X. So it's plus 28X. Remember we've got to
take extra care when we've got those minus signs.
One times by 13 is just 13. Now we need to tidy
this side up 28 takeaway 14 is just 1435 - 35 X
plus 28X. Or what's the difference there? It is 7 so
it's minus Seven X equals 30. Take the 14 away from each
side. We've minus Seven X equals
minus one. And divide both sides by minus 7 - 1 divided
by minus Seven is just a 7th again fractional answer, but
not to worry.
When we looked at these, now we want to have a look at a type
of equation which occasionally causes problems. This particular
equation or kind of equation looks relatively
straightforward. Translate numbers get rather more
difficult. It can cause difficulties, so we got three
over 5 equals 6 over X
straightforward. But what do we do? Let's think about it. First
of all, in terms of fractions, this is a fraction 3/5, and it
is equal to another fraction
which is 6. Well family obviously 3/5 is the same
fraction as 6/10 and so therefore X has got to be equal
to 10. That's not going to happen with every question. It's
not going to be as easy as that.
We're going to have to juggle with the numbers, so how do we
do that? Well, again, we need a common denominator. We need a
number that will be divisible by 5 and a number that will be
divisible by X. And the obvious
choice is 5X. So let us
multiply. Both sides by 5X. So we 5X times by
3/5 equals 5X times by 6 over X and now
five goes into five once and five goes into five
once there. X goes into X once an X goes into X. Once
there, let's remember that we're multiplying by these, so we have
one times X times three. That's 3X and divided by one, so it's
still three X equals 5 1 6, which is just 30 and divided by
one, so it's still 30.
3X is equal to 30, so X must be equal to 10, which is what we
had before. Is there another way of looking at this equation?
Well, yes there is.
If two fractions are equal that way up there also equal the
other way up.
This makes it easier still because all that we need to do
now is multiply by the common denominator and we can see what
that common denominator is. It's quite clearly 6 because six
divides into six and three divides into 6.
So we had five over three is equal to X over 6, and we're
going to multiply by this common denominator of six. So six goes
into six once on each occasion, three goes into three once and
into six twice, so again X is equal to 10 two times by 5,
one times by X.
Whichever way you use.
Doesn't matter. They should come out the same. There's no reason
why they shouldn't, but you do have to be careful. The number
work can be a bit tricky sometimes. Let's have a look at
just a couple more.
Five over 3X is equal to
25 over 27.
OK. What I think I'm going to do with these is flip them over
3X over 5 is equal to 27 over 25. I can see straight away.
I've got a common denominator here of 25. Five goes into 25
exactly and so does 25. So if I do that multiplication 25 * 3 X
over 5 is equal to 25 * 27 over
25. 25 goes into itself once on each occasion.
Five goes into itself once and five goes into 25 five times. So
I have 15X5 times by three. X is equal to 27, and so X is 27 over
15. Dividing both sides by 15 and there is here a common
factor between top and bottom of three, which gives me 9 over 5,
so that's an acceptable answer because it's in its lowest
forms. Or I could write it as
one. And four fifths, which is also an acceptable answer.
Now, some of you may not like what I did there when I flipped
it over, and we might want to think, well, how would I do it
if I had to start from there. So let's tackle that in another
way. So again we five over 3X is equal to 25 over 27 this time.
We're not going to flip it over. Let's look for a common
denominator here between these two, so we want something that
3X will divide into exactly, and something that 27 will divide
into exactly, well. Three will divide into 27, so the 27, so to
speak ought to be a part of our answer. What we need is an X,
because if we had 27 X, 3X would divide into it 9 times and the
27 would just divide into it.
X times, so that's going to be our common denominator. The
thing that we are going to multiply both sides by 27 X.
So we can look at this and we can see that X divides into X
one St X device into X. Once there we can also see that three
divides into three and three divides into 27 nine times and
over here 27 goes into 27 once each time. So I have 9 * 1
* 5 and 95 S 45.
Divided by 1 * 1, which is one, so it's still 45 equals 1 times
by 25. And times by the X there 25 X. Now I need to divide
both sides by 25 so I have 45 over 25 equals X.
This is not in its lowest terms. I can divide top on
bottom by 5, giving me 9 over 5 again, which again I can
write as one and four fifths.
Let's take one final example.
This time, let's look at some fractions, but this time,
mysteriously, the X is already on the top. That's really good
for us. All we need to do is look at what's our common
denominator, well, 7.
And 49 what number divides exactly by both of these and it
will be 49. So we just need to do 49 times by 19 X
over 7 equals 49 * 57 over 4949 goes into 49 once each
time. And Seven goes into 49 Seven times.
And so I have 7 times by 19 X equals 57 and you might say,
well, hang on a minute, shouldn't you have multiplied
out that first? Well, I didn't want to. Why didn't I want to?
Well, sometimes when you play darts, your arithmetic improves
and triple 19 on a dartboard is 57. So 19 divides into 57 three
times, and I don't want to lose that relationship. So I'm going
to divide each side by 19 so 7X.
Is equal to three, which means X must be 3 over 7.
Playing darts does help with arithmetic. We finished there
with simple linear equations. The important thing in dealing
with these kinds of equations and any kind of equation is to
remember that the equal sign is a balance. What it tells you is
that what's on the left hand side is exactly equal to what's
on the right hand side. So whatever you do to one side, you
have to do to the other side, and you must follow.
The rules of arithmetic when you do it.