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How To Solve Equations With 2 Variables On Each Side
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Feb 19, 2025
We've already learned how to solve two step equations. In this lesson, we'll learn how to solve equations with variables on each side, and also learn how to solve equations that involve grouping symbols. Chapters: 00:00 Introduction 02:13 How to solve an equation with variables on each side 04:21 Solving an equation with variables and integers on each side 06:55 Working on a real world problem 13:117 Solving equations with variables and fractions on each side
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0:00
all right so we are starting section
0:05
5.2 which deals with solving equations
0:09
with two variables all right we haven't
0:13
done that yet we've all what we've done
0:15
so far is like we've done it where you
0:18
have just like X and then solve for
0:21
solve for
0:23
um solve for solve the equation and then
0:26
we only have like numbers on other side
0:29
and stuff like that we have really done
0:30
it when we have this kind of stuff okay
0:33
so here we're going to have variables on
0:35
each side of the equation which mean it
0:38
requires us to do an extra step before
0:40
we solve the equation all right so now
0:43
the question that I get from you guys is
0:45
always why and when am I going to use it
0:48
well typical example I have a word
0:50
problem here that this can be used for
0:53
example if you let's say you want you
0:55
wanted to go training right you wanted
0:57
to go to a gym okay and you wanted to
0:59
find membership somewhere so the trainer
1:01
charges a one time fee of
1:04
$60 plus $25 for challenges do all this
1:09
okay and the new fitness club charges a
1:11
yearly fee of $450 plus $10 for each
1:15
session with a trainer now they want you
1:17
to write an equation to determine for
1:19
what number of sessions will the cost be
1:22
equal so this is something that you can
1:24
do you can use this equation to solve
1:27
this problem because most people are
1:29
going to be like why I don't know this
1:30
is 450 and extend the hour cuz this is
1:32
too much so how you going to figure it
1:34
out you can use your mathematics you can
1:36
use your mathematics to figure it out
1:38
right you can't you don't have to sit
1:40
there go I need to call such and such
1:42
because he know how to do this no you
1:44
can say to yourself I remember when I in
1:46
seventh grade you learn how to set up
1:48
equations and solve it and you can solve
1:50
it this is easy stuff right and you
1:53
because of your knowledge in mathematics
1:55
you got you have an ascendance on
1:57
somebody else meaning you know how to do
1:59
it right
2:01
now before we get to this we need to
2:04
know the basics first right now in on
2:06
the board I have an equation it says
2:08
that 3x is = to x + 8
2:14
right what do you have in your hand why
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you like when your head is like looking
2:17
down what is it P trash what is that let
2:20
me see it okay put it on the desk and
2:25
keep your eyes fast on the board all
2:28
right so 3x is equal x + 8 right now is
2:32
there anybody here who can tell me how
2:34
to solve this
2:36
problem you two always talk I want
2:39
somebody else to talk I got all right
2:45
IUS
2:47
8 what are we solving
2:50
for so what would be the best thing to
2:53
subtract
2:55
them minus
2:58
what- X right why am I doing minus X is
3:02
the
3:02
question yeah because the other side has
3:06
X and that's like terms exactly did you
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hear that because the other side has the
3:10
3x and those are like terms so we want
3:13
to put all the like terms on the same
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side right that's what we want to do we
3:19
want to put all the like terms on the
3:20
same side so we're going to subtract the
3:22
X and we're going to subtract an X here
3:26
all right and now we draw a line right
3:28
you don't have to and then this cancels
3:31
out now what's 3x - 2x 2x now 2x is
3:35
equal to 8 right 3x - x is 2x three
3:38
oranges minus an orange uh two oranges
3:42
right 3x a lot of people get like oh 3x
3:44
- x I don't know because whenever we
3:47
don't have a number next to the variable
3:49
what is that one one we know that right
3:53
so now how do I solve for x from here 2x
3:56
is = to 8 / 2 right divid by two so we
4:01
get x equals what I
4:06
mean four right I don't know what I was
4:08
thinking that yeah I didn't drink my
4:10
coffee this morning anyway so right now
4:14
the second example we have what we have
4:18
variables and integers on either side of
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the equation right which means we are
4:25
turning it up and Notch it and then now
4:27
we're going to figure out how do you
4:28
solve this I have 7 y + 8 = 4 y - 10 I
4:34
want somebody else I'm somebody else
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what 7 y + 8 = 4 y -
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10
4:46
um yeah
4:50
Kenzie so what we do first uh first you
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cancel out 7 Y and 4 Y how am I going to
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do that
5:00
by
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subtracting do subract the number
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variable no the whole whatever is 4 y
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right subract 4 y subtract 4 y from that
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right we want subtract 4 y right and
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when we do that guess what this is gone
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right that's 3 y 4 Yus and then this is
5:22
3 y + 8 right y equals to -10 then what
5:28
we do from here that you cancel out
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again what do we cancel out now 3 Y what
5:35
3 y why would we cancel we can't cancel
5:38
out 3 y just yet okay yeah eight eight
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all right we going to take out eight by
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doing what- minus8 right thanks for
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helping on
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that all right take out the eight you
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see what we doing here right first we we
5:52
got rid of the four y by bring it into
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the other side to make sure the
5:56
variables are together and now we're
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going to put the numbers together
5:59
because they all also like terms right
6:02
okay so that that's going to do that now
6:04
what's -10 - 8 what's that
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equal 18 right we have8 is equal to 3 Y
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and then
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what divided by 3 right because 3 Y is 3
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* y so we're going to do what the
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opposite which is division right
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division
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so now we got y equals what 6 or
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which negative and positive right so do
6:33
you see so this is what we going to
6:34
encounter in this section now there's
6:36
going to be different wrinkles we're
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going to have we may have like fractions
6:40
and stuff like that so do not think this
6:42
is as easy as it looks all right so now
6:46
let's do uh another one here before
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before we do that though I want to work
6:51
on the word problem real quick right
6:53
let's let's let's work on this word
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problem let's set it up so you have a
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trainer that charges a one time fee of
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$60 plus $25 for each session what would
7:05
that equation look like I want somebody
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to set that first equation for me yes go
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ahead don't erase I'm not erasing okay
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so he he charges $60 plus
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25 for each session so what would the
7:20
equation look like so 6 + 25 25 what
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dollar n 25 what for each session so
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what's the session C
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x 65 + 25x that's the first equation
7:33
right let's put number one here now
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what's the second equation going to look
7:37
like a new fitness club charges a year a
7:40
yearly few of
7:42
450 plus $10 per
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session yeah yes yeah 450 10 so it's 450
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plus 10 all right 10x that's the second
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equation right now this is asking us
7:59
write an equation to determine for what
8:02
number of sessions will the cost be
8:05
equal so how do we write an equation to
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do that what do we do
8:10
here what what do you think you're going
8:12
to do to be able to solve this we want
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the session to be equal so what we have
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to
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do yeah yes um do we first add well we
8:24
haven't even set up the equation just
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yet oh we want the SE to be equal this
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is the first this is one of the gym and
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that's the second gym so we want the
8:33
session to be equal what does that mean
8:36
translate that for
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me
8:40
yeah what do you mean by equal like the
8:43
same we want to know when yeah they're
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going to be the same they want to be
8:46
equivalent yeah equivalent okay so how
8:49
would you set that equation to solve for
8:51
how many session the cost will be the
8:54
same oh um their cost is the same yeah
8:58
so 400 50 + 10 x = 60 + 25 exactly right
9:04
maybe I didn't ask the question right I
9:05
will probably saying some wrong things
9:07
there right so for the cost to be the
9:10
same this is what we want now we need to
9:13
figure out what x
9:16
value will Le the two CS to be the same
9:19
does that make sense so this is how we
9:22
we are translating what we are hearing
9:24
we are not just making up stuff right we
9:27
are actually translating what this
9:29
problem is telling us he saying for
9:31
write write an equation to determine for
9:33
what number of sessions the cost will be
9:36
the same this is the cost of the first
9:38
gym this is the cost of the second gy
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for for how many X's these will be the
9:43
same now we can solve for x right so now
9:46
let's go ahead and solve it what do we
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do subtract 10 x all right subtract 10x
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right this is gone we have
9:58
450 = 60 + 15x and then
10:03
what subract 60 subtract 60 take it
10:08
out right that's
10:11
390 =
10:14
15x divided by 15 right somebody do it
10:17
in your
10:20
calculator you like
10:24
that what's that 26 all right so that
10:27
means if you book 26 sessions in each of
10:31
the gym you're going to pay what the
10:33
same amount of money does that make
10:36
sense so for 26
10:38
sessions it doesn't matter whether you
10:40
go here or here the cost is the same you
10:46
see so if somebody was like well if you
10:47
book 26 session with us you going to be
10:49
pay less well I just used my M and I
10:52
figure with 26 sessions I'm going to be
10:54
paying the same amount of money now we
10:55
can figure out what happens if you pay
10:57
less than 25 sessions would I go with
10:59
this this gym or this gym we just have
11:01
to plug it in there and see which one
11:02
has it higher or lower right so you see
11:05
how this is useful so don't ever say to
11:08
me we're never going to use this yes you
11:10
can yes you can and yes you should
11:14
because you know how to do it right you
11:16
know the dummy and nobody's going to
11:17
take advantage of you because you know
11:19
what you're doing right I received an
11:21
email the other day I thought it was the
11:22
funniest thing they say hey now we are
11:24
selling uh this item right and you
11:28
listen listen to this and tell me
11:30
whether this is C or not they say one
11:33
for 50 Cent and two for a
11:36
dollar so they go well yeah that's our
11:39
deal what should you be saying to these
11:42
people you're ripping me off ripping me
11:46
off it's the same thing right but the
11:49
average
11:50
person the average person may not
11:53
understand that oh yeah I'm going to get
11:54
two cuz I'm like no it's the same deal
11:57
right it's the same deal cuz for 50 for
12:01
really like that's not that's that's
12:03
silly M right we should be able to
12:05
recognize that because we have learned
12:08
it okay why is it the same because 50
12:12
Cent twice is the
12:14
dollar right two for $1 is the same as
12:18
one for 50
12:20
so Gabriel you not be making fun I will
12:23
not be making fun right now so I don't
12:25
even know why you making fun on somebody
12:27
should
12:29
yeah don't do
12:32
that that's why's never
12:35
to that's why he's never going to get a
12:38
girlfriend okay all right let's do
12:40
another one that's a little hard right
12:42
you're let's do another one that's a
12:44
little hard here so we're going to see
12:46
so let's say I have
12:48
two right over
12:50
5x I'm sure you're liking this already
12:53
right yes we are - 8 equals
12:57
to 20 right
13:01
three now why do you hate it and why do
13:04
you like
13:05
it because it's sort of familiar but it
13:08
has fractions I don't like it fraction
13:13
stuff teacher all right so what do we do
13:16
here I'm not going to solve this I like
13:20
it because you're going to be the soling
13:22
that's
13:25
right what call me because I because I
13:28
talk much so i i i w take this test well
13:31
guess what I will have to give it to you
13:33
on the quick what are we doing
13:43
here all
13:45
[Music]
13:50
right right right we subtract 2 over
13:53
five right that's fantastic so now what
13:57
happen is this we have two fractions
13:59
that we have to subtract well the
14:01
unfortunate thing is this they do not
14:03
have the same denominator so we have to
14:05
find our common denominator right this
14:08
is why I'm failing math we switch units
14:10
too fast I if I finally learn the
14:12
equation it's like I go to the next one
14:14
use that same formula and actually was
14:16
like no no you can't CH that
14:20
equation also you're ined in what why
14:23
you
14:23
actually when you're actually going to
14:25
use it all right now look at this why am
14:28
I learning if I'm not
14:31
that's
14:32
enough that's enough let me finish my
14:34
problem y'all can talk about your gripes
14:37
later on so we have that right so now we
14:40
have to solve this on the side before we
14:41
even go back to our problem so now come
14:44
on denominator 20 20 so I have to
14:47
multiply this by five right this by five
14:51
and then this one by four and my four
14:54
right so now I have
14:56
15 4x - 8 over four uh 20 over yeah 20 x
15:03
oh 20
15:04
20 all right so now we get 15 - 8 that
15:08
is seven this is confusing right that's
15:12
seven so this give us 7 over 8 7 over
15:15
20x
15:18
right + 20 =
15:21
8 all right wonderful now what do we do
15:25
you add eight well if you add eight
15:28
that's going to bring it back here so
15:30
why don't we get rid of
15:32
what subtract 20
15:35
right so we got
15:37
28 = 7 20x and then what's our last step
15:45
here excuse me the reciprocal
15:53
right no man I'm not doing my taxes my
15:55
attorney's doing my
15:56
taxes that's not how that works
15:59
yall see all right so this is gone your
16:02
husband will do them your
16:05
husb hey hey hey hey hey hey that's ni s
16:10
go2 how many
16:13
times you're
16:14
pay4 * 20 my man is doing
16:18
this8 my dad is
16:21
doing children and cleaning the house
16:24
and you're the one that job do I need to
16:27
do I need to like give it do you think
16:29
before we go no I don't want to do that
16:31
so be quiet what is that an eight that's
16:34
8 what is
16:36
that
16:42
little monop is like maatic no
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