- [Instructor] In many other videos,
we've taken equations like this
and tried to quote solve for X.
What we're gonna do in this video is deepen
our understanding a little bit about what's going on
and really think about the notion of equivalence
or equivalence statements.
So let me write that down, equivalence.
Now, what do I mean by equivalence?
Well, I'll use this equation here
to essentially keep rewriting it in equivalent ways
and I'll talk a little bit more about what that means.
So one thing that I could do to essentially
write the same equivalence statement
is I could distribute this three onto the X
or onto the X plus one and then this part of it
could be rewritten as three X plus three
and then we have minus X is equal to nine.
Now, what I will say or what it might be obvious to you
on some level is this top equation
and this second equation are equivalent.
What does that mean?
It means if one of them is true for given X,
the other one will be true for that same X as well,
and vice versa and we can write other
equivalence statements.
For example, if we were to combine X terms,
if we were to take three X and then minus X,
right over there, I could rewrite that as two X
and then I have plus three is equal to nine.
Now all three of these statements are equivalent.
If there's an X where two times that X
plus three is equal to nine,
then it is also the case that three times that X plus one
minus X is equal to nine and vice versa.
If there's some X that would make this top equation true
then it's going to make this last equation true.
And we can do other equivalence preserving operations
and you've seen them before.
You could subtract three from both sides.
In general, if you are adding or subtracting the same value
from both sides, it is equivalence preserving.
If you're distributing a value like we did
in that first step that is equivalence preserving.
If you combine like terms so to speak,
that is equivalence preserving
and so here we'll do an equivalence preserving operation.
We'll subtract three from both sides
and you would get two X is equal to six
and once again, any X that satisfies this last equation
will satisfy any of the other equations and vice versa.
Any X that's satisfies any of these other equations
will satisfy this last one
and so they are all equivalent to each other.
And then another equivalence preserving operation
is to multiply or divide both sides by a non zero constant.
And so here we could divide both sides by two,
two is not zero and it's constant
and if we did that, we will get another equivalent statement
that X is equal to three.
So any X that satisfies this and there's one, X equals three
would satisfy the other ones and any X that satisfies
any of the other ones would satisfy this last one
so these are all equivalent.
So one way to think about it,
adding the same number to both sides of an equation
that is equivalence preserving.
Multiply or dividing both sides by a non-zero constant value
is equivalence preserving.
Distributing like we did in this first step
equivalence preserving, combining like terms
equivalence preserving.
Now you're probably saying,
well, what are some non equivalence preserving operations?
Well, imagine something like this,
lemme just start with something very obvious.
If I said that X is equal to two
a non equivalence preserving operation
is if I were to add or subtract or multiply or divide
only one side of this equation by a value
let's say I only added one to the left side.
Then I would have X plus one is equal to two
and it is not the case that anything that satisfies
this second equation satisfies the top equation
or vice versa.
X equals two clearly satisfies the top equation
but it doesn't satisfy the second one,
that's because we did a non equivalence
preserving operation.
Likewise, if I only multiplied the right hand side by three,
I would get X is equal to six.
Well, by only multiplying the right hand side by value
it's not the case that anything that satisfies X equals six
will satisfy X equals two, that is somewhat obvious here.
Now there's a little bit trickier scenarios,
let's say we have the equation five X is equal to six X.
Now, one temptation is, well, I wanna do the same thing
to both sides, I could just divide both sides by X,
what'll happen in that scenario?
Well, if you divide both sides by X
you could think that an equivalence statement
is that five is equal to six and you know that there's no X
for which five is equal to six.
You can't make five equal six or six equal five
and so this would somehow make you imply that okay,
there's no X that can satisfy.
If you assume that these were equivalence statements,
you'd say there's nothing that could satisfy five equal six
so maybe there's nothing that satisfies this top equation.
But this actually isn't an equivalence preserving operation
because you're actually dealing with a scenario
where X is equal to zero and you're dividing by zero.
And so you have to be very careful
when you're dividing by a variable,
especially if the variable that makes that equation true
happens to be zero.
And so in order to be clear
that you're preserving equivalence here,
the way that I would tackle it is
I would subtract five X from both sides
and if you do that, and that is an equivalence
preserving operation, you could subtract that expression
from both sides or that term from both sides
and then you would be left with zero is equal to X
and now zero equals X and five X equals six X,
these are equivalence statements.
They are equivalent equations
anything that makes this one true is going to make
that one true and anyone that makes that one true
is going to make this one true.
Now one last thing you might have heard me say,
you can multiply or divide by non-zero value
and that's going to be equivalence preserving.
And hopefully you've just got a sense
of why dividing by zero is not a good idea.
In fact, dividing by zero is always going to be
a strange thing and it's undefined,
but also multiplying by zero.
For example, if I had, let's say, actually
lemme start over here.
If I had two X is equal to six, and if I were to multiply
both sides by zero, I would get zero is equal to zero
and zero equals zero is true for any X.
Zero is always going to be equal to zero
but the problem is, is that first statement isn't true
for all X it's only true for X equals three.
So these two are not equivalent statements.
They have a different set of Xs that will satisfy them
and so you have to be very careful when you're dealing
with things that are either are zero
you can add or subtract zeros,
obviously that's not gonna change things much,
but when you multiply both sides by zero,
you can start getting things
that are not equivalence statements
and when you multiply or divide by things
that could be zero, like variables,
that also is a dangerous game to play.