- [Instructor] In this video,
we're gonna try to solve the equation
three times X plus one minus X is equal to nine.
And, like always, I encourage you to pause this video
and try to work through this on your own.
But the emphasis of this video
is to not just get to the right answer,
but to really understand what's happening
when we do algebraically valid operations
on either side of this equation.
All right, so let's begin.
So the first thing that my brain wants to do,
and we've talked about this before,
there's many different ways to often solve an equation,
but my brain wants to simplify.
And the first thing that looks a little bit hairy here
is I that three times the expression X plus one,
and so I might be able to simplify that
if I multiply three times X and three times one.
And so if I did that,
I would get three X plus three
minus X, minus X,
is equal to nine.
And so the key here is that the left-hand side
of the second equation
is equivalent to the left-hand side of the top equation,
and so if the left-hand side of the top equation
is equal to nine,
well, so the left-hand side of this bottom equation
is going to be equal to nine as well.
Another way to think about it is
if the entire top equation is true,
the entire bottom equation must be true,
and vice versa,
but let's keep going.
So what's the next thing that we could do here
for at least just trying to simplify the left-hand side?
Well, my brain immediately sees I have three Xs here
and then I wanna take away an X right over there.
So if I have three Xs and then I take away an X,
how many Xs do I have?
Well, then I have two Xs,
and then I still have that plus three,
plus three is equal to nine.
And so, once again,
this left expression is equivalent to this left expression,
which is equivalent to this top left expression.
And if any one of these equations are true,
that must mean that all of these equations are true.
But we could keep going.
Now the next thing I like to do is
isolate the X terms on one side.
So I wanna have a left side that just has this two X there.
Well, the way to do that is
I would have to subtract three from the left side.
But, as we know, anything that we add or subtract
or multiply or divide by on one side of the equation
we have to do on the other side of the equation.
So I'll subtract three right over there.
And why does that make sense?
Well, if two X plus three is truly equal to nine,
then if I take three away from two X plus three,
if I just take three away,
it's not necessarily going to be equal to nine anymore,
so I have to take three away from nine
in order for the equivalence of both sides of the equation
to be true.
And so what do I get?
Well, then three and negative three cancel out.
And on the left-hand side I'm just left with the two X.
And on the right-hand side I have
equals nine minus three is equal to six.
I keep emphasizing it
'cause really that's the point of this video,
that two X equals six is an equivalent equation
to three times X plus one minus X equals nine.
Because we've done these algebraically valid operations
we've been able to maintain the equality
and we've been able to say, look, if two X is equal to six,
then we know that three times X plus one minus X
is equal to nine,
and if this top equation is true,
then we also know that this last equation is true.
But we're now in the home stretch,
what can we do to solve for X?
Well, if we just wanna isolate an X
here on the left-hand side,
as you can imagine, we can divide the left-hand side by two.
But if we divide the left-hand side by two,
in order to maintain this equality,
we have to divide the right-hand side by two.
Remember, equation, it has the word equality in it,
or at least the first part of the word equality in it.
If we multiply, divide, add, or subtract to one side,
then we have to do on the other side.
And so we are left with X is equal to
six divided by two,
X is equal to three.
And we have solved it.
And really X equals three continues to be an equation.
And so the equation X equals three is going to be true
if this top equation is true,
and this top equation is gonna be true
if the equation X equals three is going to be true.
Now let me finish off with a little bit
of an interesting challenge for you.
If I have five X is equal to six X,
one temptation might be,
it kind of looks like this last step we had over here,
and so why don't we divide both sides by a common factor?
And so maybe we can divide both sides here by X,
'cause this is five times X and this is six times X.
And what would we get if we did that?
Let's see, we would get,
we would get five is equal to six,
which it clearly is not equal to six,
but what just happened here?
Is it the fact that five X can never be equal to six X
and we did algebraically valid operations
and we got five equals six?
Or did we do something wrong?
Pause this video and think about it.
Well, some of you might've realized
that if X is an arbitrary non-zero number and you know that,
you could divide by X.
But what if X is zero?
If X is zero, you can't divide by zero.
And so that's actually what's going on here,
because if you do algebraically valid operations,
you will actually see that this is a case
where X is going to have to be equal to zero.
So let me rewrite it.
So this is five X is equal to six X.
Even though it looks a lot like this last step
that we had in the first equation
and you're attempted to immediately
kind of knee jerk to divide,
you have to realize that I have an X term on both sides,
or X terms on both sides.
Let me combine them.
So I could subtract five X from both sides,
or I could subtract six X from both sides.
Let me subtract five X from both sides,
and then see what happens.
And we are going to get on the left-hand side
five X minus five X is zero,
and then that's equal to six X minus five X is equal to X.
So we get the solution for that original equation,
five X is equal to six X,
is indeed X equals zero.
So the big takeaway here
is to appreciate the equivalents of these equations.
If you do algebraically valid operations,
and hopefully it's starting to make some intuitive sense
why certain operations are valid
and why other operations are not valid,
but if you do valid operations,
it's really saying that each step of your solution,
each of those equations are equivalent
to the equations before it.
If one of them is true, then the others are true.
But if you do an algebraically invalid operation,
like you're dividing by X and X could be equal to zero,
then you can start running into problems.