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Section 4.4 How To Solve Equations By Multiplying Or Dividing
In our last session, we learned how to solve equations by adding or subtracting, and in this section, we'll learn how to solve equations by way of multiplication and division.
Chapters:
00:00 Introduction
00:35 Why learn it?
02:42 Solving equations by dividing
05:24 Solving a real world problem
08:05 solving equations by multiplying
10:39 Solving equations by using the reciprocal
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0:00
so now we're going to talk about solving
0:02
equations by multiplying or dividing the
0:05
last time we talked about uh solving
0:08
equation by what adding or subtracting
0:11
that's before we went our break so we're
0:13
going to continue and then guess what
0:15
happens when we finish the section we're
0:18
going to talk about how to solve
0:19
equations in general and that is going
0:21
to include adding subtracting dividing
0:25
and multiplying and then that takes us
0:26
to our next step which is sequences okay
0:30
so now why do we do this I thought about
0:32
it cuz a lot of times when we do these
0:34
sections you guys ask me am I ever going
0:36
to use this right so why do we do this
0:38
and why do we need to learn this it's
0:40
important right so let me ask you just a
0:43
random question assume that you are 14
0:46
years old this is an assumption I don't
0:48
know who is 14 years I don't really care
0:49
I just know assume that you are 14 back
0:53
okay assume that you are 14 and I'm 42
0:56
this an assumption I assume that I'm 42
0:59
right 42 now the question is this how
1:01
many times older am I than you how would
1:04
you solve that problem right if I'm if
1:07
I'm 42 and you're 14 how many times
1:10
older am I than you anybody go ahead and
1:13
try to solve it yes sir exact number 14
1:16
42id 14 right 42id 14 now before you do
1:21
that right how would you set up an
1:23
equation to solve that problem set up an
1:26
equation to solve that problem yes did
1:29
you tell it
1:32
yeah now before you do that set up to
1:34
problem right if I'm saying assume I'm
1:36
14 years old I'm 42 right so how much
1:39
how many times older am I am I than you
1:42
so first let's try and see if you can
1:43
set up an equation for that right you
1:45
got it right it's 42 / 14 so how would
1:48
you set up an equation yes 14 * x = 42
1:52
thank you 14 * x = 42 there you go 14 *
1:58
x = what 42 that's exactly what the
2:02
statement means here right if I'm if I'm
2:05
42 and you're 14 so how many times older
2:08
am I than you this is how you set that
2:10
problem up right now I want to solve for
2:13
what x right and then you do y divide by
2:18
14 which you did right so here's what we
2:20
do here now in this section we're going
2:22
to learn how to solve this kind of
2:23
equations so I have 14 * X is = 42 and
2:27
I'm solving for what x
2:30
I want to know how many times older I'm
2:33
I am than you guys right so I'm going to
2:36
do what I need to get rid of
2:38
what the 14 the 14 by way of
2:42
what by way of dividing dividing
2:46
division because this is 14 * X remember
2:49
in our last chapter we did 14 + x = 42
2:53
and then to to solve for x we do what we
2:56
subtracted the 14 right we're doing the
2:58
same process here you do the opp
2:59
opposite right what's the opposite of
3:03
multiplication division division right
3:06
so this is 14 * x = 42 so to solve for x
3:10
I'm going to do what I'm going to divide
3:12
by 14 and remember when you do one thing
3:15
on one side you have to repeat it on the
3:18
other side so therefore X is equal to
3:20
what is three right X is equal to 3 so
3:25
that's basically saying that I'm three
3:26
times your age which means you are like
3:29
this and I'm right here right you're
3:31
very young you don't know much about
3:33
life and we yes right you're still very
3:36
young babies yes sir who El hand
3:39
up you're still a baby baby right now
3:43
knowing that right how would you solve
3:45
this problem here have -54 is equal to
3:48
6X so knowing this how would you solve
3:51
for x here
3:54
somebody divide by six on both sides
3:56
divide by six on both side right we're
3:58
doing the same thing divided by 6 on
4:00
both Sid okay you see what I'm doing
4:02
here I'm dividing by six on both s side
4:04
because this is the same process it is
4:07
-54 = 6X so to find X you divide it by 6
4:11
on both side and you get what -9 is = to
4:14
X so you see that we use this kind of
4:17
stuff in life right we just don't know
4:18
that we doing it but this is actually
4:20
what is going on now I have 7 Y is equal
4:24
to 63 Gabriel how would you solve for y
4:33
yeah can think yeah think that's what I
4:36
want you to do think so 7 y = 63 right
4:39
what's the operation here 7 y means what
4:42
7 y so therefore how am I going to solve
4:45
for
4:45
y 63 63 divid by S so I'm going to
4:51
divide by seven on both s side right so
4:53
I get y equals what 9 so do you see this
4:56
is really easy okay so it's a s y = 63
5:01
I'm makeing you guys to write this down
5:04
em em write this down this is for you to
5:07
write down I know I have the video but
5:09
you need to write this down right now we
5:12
are making progress here right so
5:15
basically you can make your own rule so
5:17
if you have a multiplication to get rid
5:18
of multiplication you have to do what
5:20
you have to divide right now we have a
5:22
word problem a Drive Through Safari Zoo
5:26
charges 1250 per person for admission
5:30
right let's say you go to KX and this is
5:31
how much they charge to get there right
5:35
$12.50 now in 1 hour the park raised
5:38
about
5:38
$675 in admission fees so they want you
5:42
to write and solve an equation to find
5:44
how many people actually visited that
5:47
out so how would you set that up
5:50
Nora
5:53
$1250 div no before you do that set up
5:56
the problem first uh $1250
5:59
that's X
6:01
mhm
6:03
equals all right like this right meaning
6:06
12 * 50 * X = 675 x represent what how
6:11
many people actually went there right
6:14
that's what we trying to find now to
6:16
find x what do we
6:18
do yeah um subtract two huh I mean
6:23
divide divide 1250 why do we divide
6:25
because this is a multiplication
6:27
multiplication so therefore we do the
6:28
opposite which is is division all right
6:31
should we divide by 1250 all right what
6:35
is that
6:40
equal what's that equal you can use your
6:44
calculator remember we've passed the
6:45
stage of no using calculators now we can
6:47
use
6:49
calculators anybody got
6:53
it MH 54 54 right 54 so 54 people
6:59
actually came to that zoo that day right
7:01
so therefore it's 54 people that's a lot
7:04
in 1 hour that's a lot of
7:06
visitors just 1 hour 54 people right cuz
7:11
everybody lik Zoo they look they like
7:13
animals because they think they're cute
7:14
until they you know
7:16
like yeah what if the zoo
7:21
is like the zoo is open hours steady
7:24
rate what this a steady rate that you do
7:26
three times what yeah 50 four leads into
7:30
other map yeah which we're going to do
7:32
later right now we're going to learn how
7:34
to do what we're going to now learn how
7:37
to solve equation by uh multiplying
7:40
right we learned how to do by dividing
7:42
now we're going to learn how to do by
7:43
multiplying right now suppose I have
7:46
this I say uh wait that was M oh yeah
7:51
dividing right
7:57
so I'm trying to find the question that
8:00
I can ask but I can't find it so I'm
8:01
just going to set it up so solving by
8:03
solving by multiplying let's just do
8:05
that solving by multiplying
8:11
right so if I have
8:14
y / 4 = to 20 right and I want to solve
8:20
for
8:21
y what do I do
8:23
here yes me yes you raise your hand um
8:28
so I would um I would multiply both
8:32
sides by four you multiply both sides by
8:35
four right we doing the it's the same
8:36
process now I'm dividing so therefore to
8:39
get rid of the division I have to use
8:41
what multiplication right so that means
8:44
I want to get rid of the four I have to
8:45
multiply by four on both sides right I'm
8:50
going to multiply by four because now we
8:51
are getting rid of the division right so
8:53
therefore this and this are going to
8:55
cancel so I'm left with y equal what 80
9:00
right so basically it's just the same
9:03
process if you have an if you have
9:05
addition you want to get R an addition
9:06
you use subtraction if you have
9:07
subtraction you want to get rid of it
9:09
you use what addition and if you have
9:11
multiplication and you're trying to
9:13
solve a problem that involves the
9:14
multiplication it's an equation then you
9:17
m you divide vice versa right so that's
9:19
the same process that we use okay now
9:22
the problem comes when you have
9:23
something like this let's say I
9:27
have S = to x / -2 right and I want to
9:34
solve this what am I going to do
9:37
here
9:41
anybody somebody
9:45
yes so what we um multiply s by X by oh2
9:54
I'm soling cuz I'm solving for what x
9:57
right so that means I do what I multiply
9:58
both sides by
9:59
what -2 right even though this a
10:03
negative don't get caught up and say
10:05
it's a negative so therefore I have to
10:06
add no the number is negative it's not a
10:09
negative it's not a negative as in an
10:11
operation it's just a negative number
10:13
does that make sense right so we get rid
10:16
of that so 7 * -2 is -14 right equal x
10:20
so this is basically what we are doing
10:22
today right and now we're going to learn
10:24
another process so assume I
10:27
have um
10:31
now we're going to learn how
10:32
to do this by using what we call the
10:36
reciprocal right reciprocal so let's say
10:39
I have 3 5 x is equal to
10:44
9 right and I want to solve
10:48
it now what is different in this in this
10:51
equation here what do we have what do we
10:54
have now
10:55
uh we have what yeah me what what do we
11:00
call this number um that's a fraction we
11:03
have a fraction right we have a fraction
11:06
right and now want still solve for x now
11:09
this is not is 3 over 5 so how am I
11:10
going to get rid of the 3 over 5 what do
11:12
I do here impr excuse me improper
11:15
fraction improper um what do you mean by
11:19
improper maybe it's your definition
11:21
Excuse
11:25
me yes but how would you solve it you
11:28
have it I think I have the idea but I
11:30
want to know how you solve it you want
11:32
to get 3 over five right so how do we do
11:34
that
11:35
here yeah go keep going let me see what
11:38
do you have in mind multip by 3 over 5
11:41
divide divide okay we can do that that's
11:44
a good idea right we can divide by 3
11:46
over 5 but it's going to give us a
11:48
little bit of a complex complexity what
11:50
would be even a better way of doing
11:53
this remember we call what I just use
11:56
the term what rece CR right so I'm going
12:01
to multiply this by what
12:04
oh yeah you have it just when you flip
12:09
it yeah so we going multiply this by
12:11
what 5 5 over three right 5 over three
12:16
because guess what happens when you
12:18
multiply by the inverse you get rid of
12:20
that right you could divide it but then
12:22
what's going to happen is when you
12:23
divide you going to have to divide it
12:24
and then flip it it's just too much work
12:26
we want to be efficient right we want to
12:28
be we want to get to the answer quick so
12:30
we multiply this by what the reciprocal
12:32
yes how is reciprocal place to take the
12:35
place of
12:36
dividing how does he do it uhuh okay
12:40
we'll get to that I'll show you right so
12:41
now this is going to cancel right and
12:43
then we have x = 9 * 5 is 45 2 that's 15
12:47
right now suppose you wanted to do it
12:49
the way um Caleb said we could do that
12:52
too we could do that right you had a
12:54
good idea it's actually what we're doing
12:56
here we going to divide by what 3 over
12:58
five right we're going to divide by 3 5
13:01
and we also have to divide this by what
13:03
35 right now remember when we divide the
13:06
fraction what do we actually do we
13:07
multiply by what flip so this becomes x
13:11
= what 9 * what 5 over three so it's
13:16
basically the same thing that we're
13:17
doing right we're doing the same thing
13:20
but to burn this step you can just
13:22
multiply this by the reciprocal so that
13:24
way you don't have to divide it first
13:26
and then flip it right you want to be
13:28
efficient this is fine if you feel like
13:30
this way works better for you then by
13:32
all means use it but you can always just
13:35
flip it and multiply the number by it's
13:38
reciprocal right so let's do some more
13:41
examples so say I
13:48
have um 6 over 7x
13:55
=
13:56
24 right so how would I solve this here
13:59
6 7x =
14:02
-4 what do I do
14:08
it um what I need to get rid of is what
14:11
we need to
14:12
know fr the fraction right which is 6
14:16
over 7 so how do I get rid of it
14:19
reciprocal reciprocal right so which is
14:21
what what's the reciprocal of this
14:22
number 7 six 7 six and then I'm going to
14:25
do the same thing here 7 over what six
14:28
right and then we can multiply that
14:30
right so now I already know that this is
14:34
like 24 over one any fraction is the
14:36
number over what one right I don't want
14:38
to multiply and then divide it cuz
14:39
that's just too much work I like to be
14:42
lazy right I know that six goes into 24
14:44
how many times four four times right so
14:48
I'm left what -4 * 7 which
14:53
is come on
14:55
guys negative what 28 right so X is -28
15:00
so that's what we do in this section so
15:02
now to get better at this kind of stuff
15:04
we're going to have to do what practice
15:07
practice and practice practice like Ivon
15:10
right Alan ion doesn't want to do
15:11
practice but we got to practice get you
15:14
don't know Alan ion who Alan iers nobody
15:17
knows Alan iers
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