We often talk about mathematics being a language.
But, hang on, what does that really mean?
Is it a secret code, or can we plug it into Google Translate?
It can feel like math is some mysterious form of communication only spoken in foreign lands.
Or like you’ll never be fluent if you weren’t born knowing it.
But actually, we are fluent.
I’m speaking in English, so today mathematics is English.
If I were speaking to you in Amharic, or Korean, or French, mathematics would be Amharic, or
Korean, or French.
No decryption needed!
So together over 15 weeks we’ll explore what we’re actually saying in algebra class
and see how math sentences speak truths about the world.
Today, we’ll start with a basic question: what does it mean to solve an equation?
G’Day!
I’m James Tanton, and this is Study Hall Algebra presented by Arizona State University
and Crash Course.
What you see here is really a sentence.
It has a subject: the quantity 5.
It has a verb: equals.
And it has an object: the quantity 2 + 3.
5 equals 2 plus 3.
Stick a period on the end and BAM... that’s a complete English sentence.
It’s also an equation, which is just a statement that two things are equal.
In fact, if you read mathematics papers, you’ll see that all the formulas and equations are
written with punctuation, because they are sentences.
Over time, we’ve developed a shorthand for efficiency, and that’s part of what can
make math confusing to look at.
Here’s another complete sentence that’s an inequality,
or a statement that describes the relationship between two things that are not equal:
9 < 5 + 3, period.
Subject, verb, object.
But something’s fishy.
It’s not true!
It’s like saying, “Pizza is a fruit.”
Sometimes sentences in English are false.
Sometimes sentences in math are false.
And sometimes it’s more complicated.
Suppose I said "Xavier’s height equals six foot five."
We can't tell whether this sentence is true because we don’t know which Xavier I’m
talking about.
If I pick one, we can check whether the sentence is true, but checking every Xavier in the
world would take a super long time.
Instead, we need a strategy for finding all the solutions, or the values that make my
sentence true.
And that’s the central idea of algebra, the fundamental branch of mathematics
where we work with math sentences that contain mystery quantities called variables or unknowns.
These rules and techniques we’ll establish are used in all types of math
because they let us communicate patterns that are true no matter what numbers are involved.
And mathematics is really about finding the truth.
So solving an algebraic equation -- an algebra sentence -- is really just answering the question
“what would our variable have to be to make this math sentence true?”
Usually we use the letters x, y, and z for variables, but when you’re writing your
own equations,
you can use whatever letter or symbol makes most sense to you: h for height, n for number,
p for platypus, up to you.
Let’s look at an equation we might see in a textbook, 3 + x = 7.
Based on our intuition from all the math we’ve done before now, we can guess that the unknown
x needs to be 4 to make this sentence true.
But what if you have something like 2x + 3 = x + 5?
Now guessing is harder.
Sure, you might get lucky and guess right on your first or second try.
But for more complicated equations, you could spend hours or days or years just trying out
different values.
So like before when we were measuring Xaviers, we need some sort of system.
One approach is to take a phrase you’ll hear often in algebra – balancing an equation
– and consider it visually.
In fact, thinking about a math problem visually is often helpful, as we’ll see again and
again in this series.
So let me pull out my handy Math Goggles, and let’s look at this through X, Y, and
Z-Ray Vision!
With my Math Goggles, balancing an equation appears like a set of scales.
To keep scales balanced, what we change on one side we have to change on the other.
So let’s put our equation on the scales.
Since x is standing in for some unknown number, we’ll view it as a box.
We can’t see what’s inside it, but we know there’s two on this side and one on
the other.
Well!
It seems these must be boxes of frogs.
And we need some extra frogs to keep the scales balanced.
Now the important question is, how many frogs per box?
Assuming they’re industry-standard frogs, each will weigh exactly the same.
And we know there’s the same number of frogs in every box labeled x.
So every x has the same value, wherever it appears in the equation.
Let’s see if we can make this simpler.
Could we take anything off each side and still have the scales balance?
Well, each side has at least 3 extra frogs.
This equivalent equation, an equation that will only be true if our original equation
is also true, is already easier to look at.
And now why not take a whole box off!
We might not know how many frogs are in each box, but we do know we’re removing the same
number of frogs from each side.
Ahh.
Now it’s clear that there are two frogs in the box!
So if we plug in 2 for x in the equation 2x+3 = x+5,
we’ll get something true.
And we do!
7 on the left equals 7 on the right.
Fortunately, most algebra doesn’t involve trying to get frogs to sit still on scales,
but making things simpler is pretty much the general strategy.
We want to keep fiddling with our original equation until we get an equivalent equation
where the solution is much easier to see.
Here we made moves to simplify our equation, but if we’re feeling cheeky, we could make
it look more complicated.
We have this friendly little x = 2.
Add 7 to both sides and that will make the solution less obvious.
Then we could add 2x to both sides.
And oh, why don’t we double each side as well -- just for kicks.
We can scale each side up or down by a number as well as just adding or subtracting, as
long as we’re careful to do it on both sides.
And now, as a final flourish, let’s subtract x from each side.
Our devious ways have taken the straightforward equation x = 2 and turned it into an unfriendly
-- but equivalent! -- one.
In general, we want to make equations clearer, not more obscure.
But as long as we’re careful to keep both sides balanced,
the kernel of truth at the heart of the equation – the solutions – will always remain unchanged.
Now so far our equations have only had one solution,
but it’s possible to end up with multiple values that make our equation true, like here.
Right away we see that if w is 55, then the equation says “zero times something equals
zero,” which is true, because anything times zero is zero.
The same thing happens if w is 707.
Then the equation reads “something times zero is zero,” which is also true.
Both the values 55 and 707 make this a true sentence, but nothing else will.
Stick in any other number for w and neither term will be 0.
So our solution set -- a list of all possible values that make our equation true -- is 55
and 707.
If we want to be sneaky, there are even equations with an infinite number of solutions.
Think for a moment about which values of z make the equation z=z true.
Any value of z you like!
So the solution set is the set of all numbers.
And on the opposite end of the spectrum, what values of x make the statement x = x+1 true?
None!
So the solution set is empty.
But these situations are rare.
If you’re working with an equation that arises from a real-world scenario or from
a textbook that wants you to practice your solving skills,
probably not every number is a solution.
Textbooks also try to avoid equations with no solutions.
So if you find that your equation seems to have all the solutions or none of the solutions,
you might have found one of these rare cases in the wild…
but probably you should go back and try again from scratch, just to make sure.
And of course, our equations can look a lot more complicated when they start involving
higher powers of x.
Like 2x2 - 6x = 0.
What can we do here?
Our balancing tricks still work, so let’s try some.
Halving all the numbers on the left makes things simpler,
so we need to halve the equation on both sides like we did with our frog scales to keep it
balanced.
And it might seem weird to divide zero by another number, but that’s perfectly legal
in math -- zero cut in half is still zero.
Now we have x2 - 3x = 0.
I’m tempted to do the same dividing trick with x since both terms on the left have an
x.
But DANGER!
We can only safely scale equations up or down by constants, which are just actual numbers
and not variables.
But x is a variable or unknown.
If it ends up being 0 and we divide by x, we’ve committed some math crimes by actually
dividing by 0.
So until we’re sure what x is, we have to treat x as possibly 0-until-proven-otherwise.
And that means we can’t divide by x.
So what else can we do?
Well, if we see a common factor, or a term that divides all our pieces on one side, we
can pull it out front.
For us, x is a common factor and we can write x(x - 3) = 0.
Here we’re not dividing both sides by x, we’re just rewriting the equation.
So now we have two things multiplied together that are equalling zero.
Like before, at least one of them has to be zero.
If x is 0, the first term is 0 which makes the equation 0.
So x = 0 is a solution.
Aha!
Guilty!
Good thing we didn’t divide before.
And if we plug in 3 for x, the second term is 0, which again makes the equation 0.
So x = 3 is another solution.
If we try anything else, like x=5 or x= 92 or x=1004, we’re multiplying two nonzero
numbers and that will never be zero!
So again we have two solutions, 0 and 3.
Basically, equation solving is actually just truth hunting.
That sounds easy, but of course it’s not!
So when you’re faced with an algebra problem and feeling stuck,
it’s way better to do something (anything!) than nothing.
But what can you do?
First, it’s generally better to make math problems look simpler.
Look for what you can do to both sides that will make the equation less complicated.
Often this will be adding or subtracting a number on both sides.
Or if there are a lot of largeish numbers involved, like in 12x2 – 36x – 480 = 1200,
look for something you can divide everything by to make them smaller.
Or if all your numbers are tiny, like 0.1x2 + 0.6x + 0.9 = 0.5,
think about how you could scale your equation up to make it easier to work with.
And second, as you’re working, write enough to be able to keep track of your steps.
You’ve probably been told to “show your work” for a grade, but the most important
person to show your work to is you.
This habit will help you keep equations balanced, keep you from making a lot of mistakes,
and make it much easier to go back and find the problem when something does go wrong.
So let’s get solving!
Next time we’ll reverse our thinking and go from English to Math when we translate
word problems into equations and pictures.
Until then, cheers!
Thanks for watching Study Hall: Algebra, which is produced by Arizona State University and
the Crash Course team at Complexly.
If you liked this video and want to keep learning with us here in Study Hall, be sure to subscribe.
You can learn more about ASU and the videos produced by Crash Course in the links in the description.
See you next time!