Intuition why we divide both sides

Algebra

The why of algebra

Why we do the same thing to both sides: simple equations
Representing a relationship with a simple equation
One-Step Equation Intuition
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One step equation intuition
Adding and subtracting the same thing from both sides
Intuition why we divide both sides
Why we do the same thing to both sides: two-step equations
Why we do the same thing to both sides: multi-step equations
Why we do the same thing to both sides basic systems

Intuition why we divide both sides Conceptual understanding of why one needs to divide both sides of an equation to solve for a variable

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Intuition why we divide both sides

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We have our scale again, with some masses on the left-hand side,
and some masses on the right-hand side and we see that our scale is balanced.
We have the same total mass on the left-hand side that we have on the right-hand side.
Instead of labeling the mystery masses with a question mark,
I have labeled them all "x."
Since they all have an 'x' on them, we know that each of these has the same mass.
What I am curious about is: what is that mass?
What is the mass of each of these mystery masses?
I will let you think about this for a second: how would you figure out what the mystery mass is,
or what this x-value actually is? How many kilograms is each of these things?
What could you do to either one or both sides of this scale?
I will give you a few seconds to think about that...
You might be tempted to say,
"If I could end up with just one mystery mass, and if I keep my scale balanced,
then that thing will be equal to whatever is on the right-hand side."
And that part would be a true statement.
And to get to just one of these mystery masses, you might say,
"Why don't I just remove two of them?
Why don't I just remove that one and that one?
Then I will just be left with that single one."
But if you just removed these two,
the left-hand side would become lighter than the right-hand side,
So left will move up, and right will move down.
Now you could say: "OK, I understand! Whatever I do to the left-hand side,
I have to do to the right-hand side so I can keep my scale balanced."
So you might say: "Why don't I remove 2 of these mystery masses from the right-hand side?"
But that is a problem too, because you don't know what this mystery mass is.
You could try to remove two,
but how many of these blocks represent a mystery mass?
We actually do not know.
But you might then say, "I have got three of these things here,
If I multiply what I have here by a third,
If I leave a third of the stuff here, and I leave a third of the stuff here,
then the scales should be balanced."
If this has the same total mass as this,
then 1/3 of this total mass will be equal to 1/3 of that total mass.
So let's just keep only one-third of this here.
This is equivalent to multiplying by one-third.
If we are only going to keep 1/3 there, we will be left with only one of the masses,
And if we only keep 1/3 here, let's see,
we have 1, 2, 3, 4, 5, 6, 7, 8, 9 masses.
If we multiply this by one-third: 1/3 x 9 is only 3.
So we remove these, and now we have 1/3 of what we had on the right-hand side,
and 1/3 of what we originally had on the left-hand side.
And they will be balanced, because we took 1/3 of the same total masses.
And so, what you are left with is just one of these mystery masses
- this 'x' thing over here, whatever x might be -
and you have 3 kilograms on the right-hand side.
So you can make the conclusion, as you kept this thing balanced the whole time,
that x = 3.

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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...

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At 2:33, Sal says "double bonds" but should say "single bonds."

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