One-Step Equation Intuition

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
One step equation intuition exercise intro
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
Why all the letters in Algebra?

One-Step Equation Intuition Why we solve one-step equations the way we do.

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One-Step Equation Intuition

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Once again we have three identical objects
they all have the same mass, but we don't know what the mass is.
But we do know that if you total up their mass,
it is the same exact mass as these nine objects.
and each of these 9 masses have a mass of exactly 1 kilogram.
So we have a total of 9 kg on this side,
and over here we have 3 objects that all have the same mass
but we don't know what that mass is, we are just calling that mass 'X'.
What I want to do is to tackle this a bit more symbolically.
In the last video, we said "hey, why don't we just multiply 1/3 of this and 1/3 of that?"
and then we would keep things balanced.
keeping 1/3 of the same mass, so this total is the same as this one.
and that is why the scale is balanced.
Now, let's think about how we can represent this symbolically.
The first thing I want you to think about is:
Can we set up an equation
that expresses these 3 things of mass 'X'
and that in total their mass is equal to the total mass over here.
Can we express that as an equation?
I will give you a few seconds to do it...
[silence]
Let's think about it. Over here, we have 3 thing with mass 'X'
so their total mass we could write as X + X + X
And over here, we have nine things with a mass of one kilogram,
which we could write as 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.
And actually, this is a mathematical representation
We set it up as an equation, so it is an algebraic representation.
It is not the simplest possible way to do it, but it is reasonable.
I we want, we could say: "If have an X plus another X plus another X, I have 3 X's."
So I could re-write this as 3X and 3X will be equal to...
--If I sum up all these ones over here, we have 9 of them,
So I have 3X = 9.
Let me make sure that is right [counts to 9]
So that is how we would set up.
The next question is: What could we do mathematically
to either one of these equations (but we will focus on this one)
What could we do mathematically in order to solve for the X
Or to figure out what that mystery mass actually is?
I will give you another second or two to think about it.
[silence]
When we did it the last time, with the scales,
we said "We have 3 of these X's here. We want to have just 1 X here,"
So we could say "whatever this X is, if the scales stay balanced,
It will be the same as what we have there."
There might be a temptation to subtract 2 of the X's from this side,
but that will not help us, we can even see that mathematically over here,
if we subtracted 2 X's from both sides,
on the left-hand side we would have 3x - 2x, and on the right you would have 9 - 2x
And you would be left with:
3 of something minus 2 of something is 1 of something
so you would have 1x there, if you get rid of 2 of them,
but on the right-hand side, you will get 9 minus 2 X's.
So the X's still don't help you out, you still have this mystery mass on the right-hand side.
That doesn't help, so instead what we say is - and we did this the last time -
We said "what if we took a third of these things?"
If we take 1/3 of these things and we take 1/3 of those things,
we should still get the same mass on both sides because the original things had the same mass.
The equivalent of doing this mathematically is to say
"Why don't we multiply both sides by one-third?"
Or another way to say it is "we can divide both sides by three"
Multiplying by 1/3 is the same thing as dividing by 3.
So we multiply both sides by one-third.
When you multiply by one-third, visually, over here, if you had 3 X's
and you multiplied by 1/3, you would have only one X left.
If you have 9 of these 1 kg boxes, and you multiply by 1/3, you will have 3 left.
Over here you can even visually divide by 3 - which is the same as multiplying by 1/3 -
-- so you divide by 3 - you have an X is equal to 1 + 1 + 1. X = 3.
Or you see here that X = 3.
Over here you do the math: 1/3 * 3 = 1 and you're left with one X
so you are left with X = 9 * 1/3, or you can do 9 divided by 3,
which is equal to 3.

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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

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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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