>> Hi, this is Julie Harland
and I'm Your Math Gal.
Please visit my web site
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
In this video, we show how
to determine whether a number
is a solution
of a particular equation
and we do the following
6 problems.
We're gonna be working
with equations and try
to determine
if a solution given is
really correct.
So first of all,
an equation has an equal sign,
there is a left side
and a right side separated
by the-- this little equal
symbol here
and if it's a true equation,
then the expression
on the left side must equal
to the expression
of the right side.
So if you look
at this first 1.
On the left side,
we have 2 plus 5
and on the right side we
have 7.
Since 2 plus 5 is 7,
then certainly the left side
equals the right side
and so this is true.
This is a true equation.
Now, let's look at the next 1,
we have nine minus 1 equal 3,
nine minus 1 is 8.
So it's not equal to 3.
So this would be a false
equation, okay?
Now, you could instead
of writing 9 minus 1 equals 3
is false, you could just slash
right through this
and that says 9 minus 1 is not
equal to 3, okay?
Now, when an equation contains
variables the solutions
to that equation are the
values of the variables
that make the equation true.
In other words, the left side
and right side equals the same
number once you put--
plug in those numbers
for the variables.
Now, an equation might have no
solutions at all,
it might have 1 solution,
it could have 2, 3,
4 solutions
or it might have infinitely
many solutions.
When we solve equations,
we'll figure out how
to find what the solutions are
but for now,
let's just take this example X
minus 3 equals 8.
This is an equation.
And so the question is this.
Is 10 a solution?
So, we're just going
to be given the question is 10
a solution
to this particular equation
and we're gonna answer yes
or no.
So, you could probably do this
in your head, If you put in 10
for the variable which is X,
10 minus3 we know is 7,
it's not gonna equal 8.
But in general, this is what I
like to do.
I like to underline the entire
equation and you put--
draw a line
under the equal sign.
So it's like a T might--
the left side
of this horizon line might be
more or less
than the right side.
And then we're gonna plug
in that number for X
on 1 side.
So, I have 10 minus 3
and we're simply going
to simplify that expression,
10 minus 3 is 7.
Then we're gonna do the same
thing on the right,
we're gonna plug in 10
if there is a variable
but there is no variable here
so there is nothing
to plug in.
So if there is nothing plug
in we'll simply write
that number which is 8
and we're gonna compare.
When I plugged in that number,
did I get the same number
on both sides because if so,
then I would have a true
equation because when I
plugged in that number,
the left side would equal the
right side.
But as you can see the left
side is 7 the right side is 8
when I plugged in 10 for X.
So it does not give the same
value on both sides.
The answer
to this question is no.
Alright, how
about you try this one,
is 5 the solution
of 2X minus 1 equal
to X plus 4?
So you wanna right
down the equation,
you want to underline the
entire thing and then
under the equal sign,
you're gonna do this.
So we're simply going
to simplify the left hand side
until we get a single number
and then we're gonna simplify
the right hand side
until we get a single number
after plugging in 5
for the variables.
So go ahead and try that.
Okay, so let's do it.
So I have 2X.
Remember that means 2 times X
so that means 2 times,
let's say I'm gonna put
in the 5 minus 1
and we're simply going
to simplify this expression
on the left using the order
of operations.
So that will give me a 10
minus 1 which is nine.
So the left side
of equation is 9 when you put
in 5, now let's do the
right side.
We're gonna put in 5 for X
and then we're gonna simplify
that and I get 9.
So I got the same number
on both sides
when I simplified it.
So is 5 a solution?
Well, when you plugged in 5,
the left side was equal
to the right hand side
so the answer
to this question is yes.
Now, the directions in a book
or in a test may be written a
little bit differently,
something like this:
Decide if 5 is a solution
of the equation 2X minus 1
equals X plus 4.
So, instead
of them asking you a question
which is a yes or no answer,
they're asking you
to just decide whether it's
a solution.
You would do the problem
exactly the same way.
Up here, you'd right the
equation, simplify the left
hand side,
simplify the right hand side
then you would see yes it's a
solution so you could say yes,
it's a solution.
That would be fine or you--
[ Pause ]
>> Alright,
here is another problem,
is 6 a solution
of tow-thirds X equals X
minus 2.
Alright, go ahead
and try this problem
by putting the video
on pause doing it
and then turning it back
on again.
Right, so we're going
to put it 6 per X. Now,
since I have a fraction times
X, two-thirds times X,
you can write times 6
of if you'd
like you could write it times
6 over 1, some people
like to write fractions
so they foot have a numerator
and denominator.
So let's go ahead
and simplify this.
So let's see,
conveniently 3 goes
into 6 twice so I have
in the numerator 2 times 2
so that's 4 over 1 and then 4
over 1 means 4 divided by 1
which is 4.
On the left hand side,
we're gonna plugged in 6 for X
and we're gonna simplify that
and I also got 4.
So when I plugged in 6,
I got a 4 on both sides.
That does not mean
that 4 is the solution,
it means that 6 was the
correct number to plug
in for X. That's the 1
that made it true.
So the answer
to this question is also yes.
Here is the next 1 for you
to try on your own,
put the video on pause
and try this.
Alright, so let's start
by underlining the entire
equation and drawing a
vertical line and plugging
in 4 for the variable
if there is a variable
on the left hand side.
So, we've got to write the X,
will put in the 4
and we copy everything else,
just putting a 4 in place
of everywhere there is an X
and now we're gonna do order
of operation.
So we have
to do inside the parentheses
first which is the subtraction
4 minus 2 so I can copy this
for in the outside
and then inside the
parenthesis,
4 minus 2 is simply 2.
Now what does this mean?
Four next to a parenthesis
that means 4 times 2
so the left hand side is 8,
right?
Now let's do the right
hand side.
We're gonna put in 4 for X
so I'm gonna--
you don't have to do this
but it's a good idea
to put your variable
in parentheses
if we're gonna raise it
to some power
so I'm gonna put 4 squared
like that but you could just
write 4 squared
without the parenthesis
around the 4,
this just make it clear what
you're squaring.
So what does 4 square means?
That means 4 times 4
or 16 plus 5, that's 21
and so the left hand side
simplified to 8
and the right hand side
simplified to 21
so 4 must not be a solution.
So the answer is no,
it's not a solution.
So it's okay by the way
to write this
as 4 squared plus 5 as long
as this number here is
positive number,
it's okay to do it
without parentheses
but when you start working
with negative numbers it will
be imperative
that you do put it
in parentheses.
That's what's coming up.
Here is another one for you
to try on your own first.
Right, so let's make the T
by underlining
and we're gonna put in 2
for X. So what does this mean
2 times X squared?
Well, I'm gonna write
that out, it means 2 times X
squared and I'm putting in a 2
for my X and again I'm gonna
put that 2
in parenthesis although you
don't have to--
if it's a positive number you
don't have to
but later you'll see why it's
a good practice to go ahead
and do it now.
>> And then I'm gonna do order
of operations
which means exponent is an X,
I'm gonna have
to do this 2 squared
so I write down this first 2
and then 2 squared is 4 then I
just copy the rest
of the problem.
Alright, next to my order
of operations is
multiplication comes next
so I have 2 times 4, 8 minus 1
and that's 7.
Now, on the right hand side,
I'm gonna plug in 2 for X,
4X means 4 times 2 minus 1
so I have 8 minus 1 which is 7
so both sides simplified to 7
so that tells me 2 must be the
correct solution.
So yes, 2 is a solution.
Now, notice the way I'm
writing this,
I'm not saying it's the only
solution but it's one
of the solutions.
There might be another
solution as well
and we'll be solving equations
later but for now we're just
checking to see
if a particular number is
at least 1 of the solutions.
Let's do the same problem
by rewriting this
to 2X squared minus 1 a little
bit differently before we plug
in the number for X.
So remember,
2 times X squared means 2
times X times X minus 1
and this means 4 times X
minus 1.
If you want you could write it
out this way
and then plug in the 2.
So we do that on the right
hand side and then I
immediately get
to do the multiplication.
What you've done is eliminate
doing any problems
with exponents.
So you're rewriting the
exponents as a multiplication
so I have now 8 minus 1 are 7
and now we're gonna--
the right hand side is exactly
the same as it was previously.
We're just gonna put in the 2
for X and again we're gonna do
multiplication first,
8 minus 1 is 7.
So, it would still get the
same answer here,
the answer would be yes it's
a solution.
Let's do 1 more problem,
1 that has fractions in it,
try this on your own first
by putting the video on pause.
Okay let's do it, 8 times X,
well I'm gonna multiply
that by a fractions
so I'm gonna write 8
as 8 over 1.
That's optional
and I'm gonna put
in three-fourths for X
and I'm gonna simplify
that left hand side.
So, let's see, can I cancel
when I do this multiplication?
Yes. So, what do I get
with this multiplication,
I have 2 times 3 which is 6
over 1 so that's going
to just give me a 6 'cause 6
divided by 1 is 6
and 6 minus 1 is 5.
Now, on the right hand side,
I'm also gonna do 12 times an
X but I'm gonna plug
in for X. So,
I'm gonna write 12
as 12 over 1.
Again, optionally you could
just right 12
if you want times 3/4
and copy plus 3.
One thing a lot
of students forget
to do is copy the whole rest
of the problem
when they start plugging
something in and simplifying.
Every line should be equal
to the line above it.
Alright, so can I simplify?
Yes. I can cancel 4
and the 12 goes 3 times
so I have in the numerator 3
times 3 which is 9
and the denominator 1 times 1,
9 divided by 1 is 9
and 9 plus 3 is 12
so this time I did not get the
same number on both sides
so the answer is no,
three-fourths is not
a solution.
[ Silence ]