0:00:00.650,0:00:12.390
Let's say we have the equation[br]7 times x is equal to 14.

0:00:12.390,0:00:15.970
Now before even trying to solve[br]this equation, what I want to

0:00:15.970,0:00:20.080
do is think a little bit about[br]what this actually means.

0:00:20.080,0:00:26.870
7x equals 14, this is the exact[br]same thing as saying 7 times x

0:00:26.870,0:00:35.000
-- let me write it this way --[br]7 times x -- we'll do the x in

0:00:35.000,0:00:41.270
orange again -- 7 times[br]x is equal to 14.

0:00:41.270,0:00:42.860
Now you might be able to[br]do this in your head.

0:00:42.860,0:00:45.490
You could literally go[br]through the 7 times table.

0:00:45.490,0:00:48.960
You say well 7 times 1 is equal[br]to 7, so that won't work.

0:00:48.960,0:00:54.350
7 times 2 is equal to[br]14, so 2 works here.

0:00:54.350,0:00:55.920
So you would immediately[br]be able to solve it.

0:00:55.920,0:00:58.510
You would immediately, just[br]by trying different numbers

0:00:58.510,0:01:01.220
out, say hey, that's[br]going to be a 2.

0:01:01.220,0:01:03.200
But what we're going to do in[br]this video is to think about

0:01:03.200,0:01:04.780
how to solve this[br]systematically.

0:01:04.780,0:01:07.100
Because what we're going to[br]find is as these equations get

0:01:07.100,0:01:09.680
more and more complicated,[br]you're not going to be able to

0:01:09.680,0:01:11.670
just think about it and[br]do it in your head.

0:01:11.670,0:01:14.590
So it's really important that[br]one, you understand how to

0:01:14.590,0:01:16.510
manipulate these equations,[br]but even more important to

0:01:16.510,0:01:18.280
understand what they[br]actually represent.

0:01:18.280,0:01:22.090
This literally just says 7[br]times x is equal to 14.

0:01:22.090,0:01:24.530
In algebra we don't[br]write the times there.

0:01:24.530,0:01:27.620
When you write two numbers next[br]to each other or a number next

0:01:27.620,0:01:30.280
to a variable like this, it[br]just means that you

0:01:30.280,0:01:31.100
are multiplying.

0:01:31.100,0:01:33.920
It's just a shorthand,[br]a shorthand notation.

0:01:33.920,0:01:37.040
And in general we don't use the[br]multiplication sign because

0:01:37.040,0:01:40.970
it's confusing, because x is[br]the most common variable

0:01:40.970,0:01:41.840
used in algebra.

0:01:41.840,0:01:49.330
And if I were to write 7 times[br]x is equal to 14, if I write my

0:01:49.330,0:01:52.140
times sign or my x a little[br]bit strange, it might look

0:01:52.140,0:01:54.150
like xx or times times.

0:01:54.150,0:01:56.410
So in general when you're[br]dealing with equations,

0:01:56.410,0:01:58.910
especially when one of the[br]variables is an x, you

0:01:58.910,0:02:01.350
wouldn't use the traditional[br]multiplication sign.

0:02:01.350,0:02:05.020
You might use something like[br]this -- you might use dot to

0:02:05.020,0:02:06.130
represent multiplication.

0:02:06.130,0:02:10.040
So you might have 7[br]times is equal to 14.

0:02:10.040,0:02:11.720
But this is still[br]a little unusual.

0:02:11.720,0:02:13.750
If you have something[br]multiplying by a variable

0:02:13.750,0:02:15.080
you'll just write 7x.

0:02:15.080,0:02:18.210
That literally means 7 times x.

0:02:18.210,0:02:22.750
Now, to understand how you can[br]manipulate this equation to

0:02:22.750,0:02:25.040
solve it, let's visualize this.

0:02:25.040,0:02:27.710
So 7 times x, what is that?

0:02:27.710,0:02:29.690
That's the same thing -- so I'm[br]just going to re-write this

0:02:29.690,0:02:32.720
equation, but I'm going to[br]re-write it in visual form.

0:02:32.720,0:02:34.880
So 7 times x.

0:02:34.880,0:02:37.740
So that literally means x[br]added to itself 7 times.

0:02:37.740,0:02:40.230
That's the definition[br]of multiplication.

0:02:40.230,0:02:47.900
So it's literally x plus x plus[br]x plus x plus x -- let's see,

0:02:47.900,0:02:51.830
that's 5 x's -- plus x plus x.

0:02:51.830,0:02:54.660
So that right there[br]is literally 7 x's.

0:02:54.660,0:02:56.300
This is 7x right there.

0:02:56.300,0:02:57.270
Let me re-write it down.

0:02:57.270,0:03:03.030
This right here is 7x.

0:03:03.030,0:03:07.610
Now this equation tells us[br]that 7x is equal to 14.

0:03:07.610,0:03:10.800
So just saying that[br]this is equal to 14.

0:03:10.800,0:03:13.300
Let me draw 14 objects here.

0:03:13.300,0:03:19.640
So let's say I have 1,[br]2, 3, 4, 5, 6, 7, 8,

0:03:19.640,0:03:23.640
9, 10, 11, 12, 13, 14.

0:03:23.640,0:03:26.790
So literally we're saying[br]7x is equal to 14 things.

0:03:26.790,0:03:29.070
These are equivalent[br]statements.

0:03:29.070,0:03:32.280
Now the reason why I drew[br]it out this way is so that

0:03:32.280,0:03:34.950
you really understand what[br]we're going to do when we

0:03:34.950,0:03:37.110
divide both sides by 7.

0:03:37.110,0:03:39.820
So let me erase[br]this right here.

0:03:39.820,0:03:44.175
So the standard step whenever[br]-- I didn't want to do that,

0:03:44.175,0:03:48.720
let me do this, let me[br]draw that last circle.

0:03:48.720,0:03:52.360
So in general, whenever you[br]simplify an equation down to a

0:03:52.360,0:03:55.080
-- a coefficient is just the[br]number multiplying

0:03:55.080,0:03:55.750
the variable.

0:03:55.750,0:03:58.465
So some number multiplying the[br]variable or we could call that

0:03:58.465,0:04:00.990
the coefficient times a[br]variable equal to

0:04:00.990,0:04:01.990
something else.

0:04:01.990,0:04:05.210
What you want to do is just[br]divide both sides by 7 in

0:04:05.210,0:04:08.240
this case, or divide both[br]sides by the coefficient.

0:04:08.240,0:04:12.130
So if you divide both sides[br]by 7, what do you get?

0:04:12.130,0:04:16.150
7 times something divided[br]by 7 is just going to be

0:04:16.150,0:04:17.690
that original something.

0:04:17.690,0:04:22.350
7's cancel out and 14[br]divided by 7 is 2.

0:04:22.350,0:04:26.180
So your solution is going[br]to be x is equal to 2.

0:04:26.180,0:04:29.200
But just to make it very[br]tangible in your head, what's

0:04:29.200,0:04:32.570
going on here is when we're[br]dividing both sides of the

0:04:32.570,0:04:36.260
equation by 7, we're literally[br]dividing both sides by 7.

0:04:36.260,0:04:37.250
This is an equation.

0:04:37.250,0:04:39.850
It's saying that this[br]is equal to that.

0:04:39.850,0:04:43.230
Anything I do to the left hand[br]side I have to do to the right.

0:04:43.230,0:04:46.130
If they start off being equal,[br]I can't just do an operation

0:04:46.130,0:04:48.970
to one side and have[br]it still be equal.

0:04:48.970,0:04:50.100
They were the same thing.

0:04:50.100,0:04:53.420
So if I divide the left hand[br]side by 7, so let me divide

0:04:53.420,0:04:55.150
it into seven groups.

0:04:55.150,0:04:58.920
So there are seven x's here,[br]so that's one, two, three,

0:04:58.920,0:05:01.690
four, five, six, seven.

0:05:01.690,0:05:04.780
So it's one, two, three, four,[br]five, six, seven groups.

0:05:04.780,0:05:07.240
Now if I divide that into[br]seven groups, I'll also want

0:05:07.240,0:05:11.350
to divide the right hand[br]side into seven groups.

0:05:11.350,0:05:16.530
One, two, three, four,[br]five, six, seven.

0:05:16.530,0:05:20.370
So if this whole thing is equal[br]to this whole thing, then each

0:05:20.370,0:05:25.660
of these little chunks that we[br]broke into, these seven chunks,

0:05:25.660,0:05:27.860
are going to be equivalent.

0:05:27.860,0:05:32.070
So this chunk you could say[br]is equal to that chunk.

0:05:32.070,0:05:34.530
This chunk is equal to[br]this chunk -- they're

0:05:34.530,0:05:35.490
all equivalent chunks.

0:05:35.490,0:05:37.360
There are seven chunks[br]here, seven chunks here.

0:05:37.360,0:05:41.390
So each x must be equal[br]to two of these objects.

0:05:41.390,0:05:46.240
So we get x is equal to, in[br]this case -- in this case

0:05:46.240,0:05:48.830
we had the objects drawn[br]out where there's two of

0:05:48.830,0:05:50.680
them. x is equal to 2.

0:05:50.680,0:05:53.210
Now, let's just do a couple[br]more examples here just so it

0:05:53.210,0:05:55.565
really gets in your mind that[br]we're dealing with an equation,

0:05:55.565,0:05:58.320
and any operation that you do[br]on one side of the equation

0:05:58.320,0:06:00.540
you should do to the other.

0:06:00.540,0:06:04.040
So let me scroll[br]down a little bit.

0:06:04.040,0:06:13.550
So let's say I have I say[br]I have 3x is equal to 15.

0:06:13.550,0:06:15.640
Now once again, you might be[br]able to do is in your head.

0:06:15.640,0:06:17.780
You're saying this is[br]saying 3 times some

0:06:17.780,0:06:19.490
number is equal to 15.

0:06:19.490,0:06:22.380
You could go through your 3[br]times tables and figure it out.

0:06:22.380,0:06:25.150
But if you just wanted to do[br]this systematically, and it

0:06:25.150,0:06:27.680
is good to understand it[br]systematically, say OK, this

0:06:27.680,0:06:30.500
thing on the left is equal[br]to this thing on the right.

0:06:30.500,0:06:31.995
What do I have to do to[br]this thing on the left

0:06:31.995,0:06:33.640
to have just an x there?

0:06:33.640,0:06:36.690
Well to have just an x there,[br]I want to divide it by 3.

0:06:36.690,0:06:39.930
And my whole motivation for[br]doing that is that 3 times

0:06:39.930,0:06:43.620
something divided by 3, the 3's[br]will cancel out and I'm just

0:06:43.620,0:06:45.120
going to be left with an x.

0:06:45.120,0:06:48.190
Now, 3x was equal to 15.

0:06:48.190,0:06:52.240
If I'm dividing the left side[br]by 3, in order for the equality

0:06:52.240,0:06:57.230
to still hold, I also have to[br]divide the right side by 3.

0:06:57.230,0:06:58.560
Now what does that give us?

0:06:58.560,0:07:01.070
Well the left hand side, we're[br]just going to be left with

0:07:01.070,0:07:04.130
an x, so it's just[br]going to be an x.

0:07:04.130,0:07:08.040
And then the right hand side,[br]what is 15 divided by 3?

0:07:08.040,0:07:10.760
Well it is just 5.

0:07:10.760,0:07:13.570
Now you could also done this[br]equation in a slightly

0:07:13.570,0:07:16.140
different way, although they[br]are really equivalent.

0:07:16.140,0:07:21.160
If I start with 3x is equal to[br]15, you might say hey, Sal,

0:07:21.160,0:07:25.260
instead of dividing by 3, I[br]could also get rid of this 3, I

0:07:25.260,0:07:28.110
could just be left with an x if[br]I multiply both sides of

0:07:28.110,0:07:30.670
this equation by 1/3.

0:07:30.670,0:07:34.710
So if I multiply both sides[br]of this equation by 1/3

0:07:34.710,0:07:35.830
that should also work.

0:07:35.830,0:07:38.980
You say look, 1/3 of 3 is 1.

0:07:38.980,0:07:41.710
When you just multiply this[br]part right here, 1/3 times

0:07:41.710,0:07:46.170
3, that is just 1, 1x.

0:07:46.170,0:07:52.130
1x is equal to 15 times[br]1/3 third is equal to 5.

0:07:52.130,0:07:56.140
And 1 times x is the same thing[br]as just x, so this is the same

0:07:56.140,0:07:58.400
thing as x is equal to 5.

0:07:58.400,0:08:01.660
And these are actually[br]equivalent ways of doing it.

0:08:01.660,0:08:05.180
If you divide both sides by[br]3, that is equivalent to

0:08:05.180,0:08:10.340
multiplying both sides[br]of the equation by 1/3.

0:08:10.340,0:08:12.160
Now let's do one more and I'm[br]going to make it a little

0:08:12.160,0:08:15.270
bit more complicated.

0:08:15.270,0:08:16.840
And I'm going to change the[br]variable a little bit.

0:08:16.840,0:08:36.200
So let's say I have 2y[br]plus 4y is equal to 18.

0:08:36.200,0:08:37.956
Now all of a sudden it's[br]a little harder to

0:08:37.956,0:08:38.540
do it in your head.

0:08:38.540,0:08:42.420
We're saying 2 times something[br]plus 4 times that same

0:08:42.420,0:08:45.260
something is going[br]to be equal to 18.

0:08:45.260,0:08:47.400
So it's harder to think[br]about what number that is.

0:08:47.400,0:08:48.110
You could try them.

0:08:48.110,0:08:51.910
Say if y was 1, it'd be 2[br]times 1 plus 4 times 1,

0:08:51.910,0:08:52.710
well that doesn't work.

0:08:52.710,0:08:54.390
But let's think about how[br]to do it systematically.

0:08:54.390,0:08:56.250
You could keep guessing and[br]you might eventually get

0:08:56.250,0:08:58.840
the answer, but how do you[br]do this systematically.

0:08:58.840,0:08:59.830
Let's visualize it.

0:08:59.830,0:09:04.540
So if I have two y's,[br]what does that mean?

0:09:04.540,0:09:08.800
It literally means I have two[br]y's added to each other.

0:09:08.800,0:09:11.990
So it's literally y plus y.

0:09:11.990,0:09:15.300
And then to that I'm[br]adding four y's.

0:09:15.300,0:09:18.350
To that I'm heading four y's,[br]which are literally four

0:09:18.350,0:09:20.090
y's added to each other.

0:09:20.090,0:09:25.220
So it's y plus y plus y plus y.

0:09:25.220,0:09:28.630
And that has got to[br]be equal to 18.

0:09:28.630,0:09:34.530
So that is equal to 18.

0:09:34.530,0:09:38.840
Now, how many y's do I have[br]here on the left hand side?

0:09:38.840,0:09:41.520
How many y's do I have?

0:09:41.520,0:09:45.590
I have one, two, three,[br]four, five, six y's.

0:09:45.590,0:09:49.070
So you could simplify this[br]as 6y is equal to 18.

0:09:49.070,0:09:50.930
And if you think about it[br]it makes complete sense.

0:09:50.930,0:09:57.350
So this thing right here,[br]the 2y plus the 4y is 6y.

0:09:57.350,0:10:00.320
So 2y plus 4y is 6y,[br]which makes sense.

0:10:00.320,0:10:02.840
If I have 2 apples plus[br]4 apples, I'm going

0:10:02.840,0:10:03.850
to have 6 apples.

0:10:03.850,0:10:07.810
If I have 2 y's plus 4 y's[br]I'm going to have 6 y's.

0:10:07.810,0:10:09.615
Now that's going to[br]be equal to 18.

0:10:12.270,0:10:15.250
And now, hopefully, we[br]understand how to do this.

0:10:15.250,0:10:18.490
If I have 6 times something is[br]equal to 18, if I divide both

0:10:18.490,0:10:22.710
sides of this equation by 6,[br]I'll solve for the something.

0:10:22.710,0:10:29.920
So divide the left hand[br]side by 6, and divide the

0:10:29.920,0:10:31.870
right hand side by 6.

0:10:31.870,0:10:39.100
And we are left with[br]y is equal to 3.

0:10:39.100,0:10:40.160
And you could try it out.

0:10:40.160,0:10:41.750
That's what's cool[br]about an equation.

0:10:41.750,0:10:44.290
You can always check to see[br]if you got the right answer.

0:10:44.290,0:10:45.220
Let's see if that works.

0:10:45.220,0:10:52.120
2 times 3 plus 4 times[br]3 is equal to what?

0:10:52.120,0:10:56.640
2 times 3, this[br]right here is 6.

0:10:56.640,0:10:59.580
And then 4 times 3 is 12.

0:10:59.580,0:11:03.410
6 plus 12 is, indeed,[br]equal to 18.

0:11:03.410,0:11:05.640
So it works out.