Hi, I’m Rob. Welcome to Math Antics.
In our last Algebra video, we learned that Algebra involves equations
that have variables or unknown values in them.
And we learned that solving an equation means figuring out what those unknown values are.
In this video, we’re going to learn how to solve some very simple
Algebraic equations that just involve addition and subtraction.
Then in the next video, we’ll learn how to solve some simple equations involving multiplication and division.
Are you ready?… I thought so!
Okay… so if you’ve got an equation that has an unknown value in it,
then the key strategy for solving it is to rearrange the equation
until you have the unknown value all by itself on one side of the equal sign,
and all of the known numbers on the other side of the equal sign.
Then, you’ll know just what the unknown value is.
But, how do we do that? How do we rearrange equations?
Well, we know that Algebra still uses the four main arithmetic operations
(addition, subtraction, multiplication and division)
and we can use those operations to rearrange equations,
as long as we understand one really important thing first.
We need to understand that an equation is like a balance scale.
You’ve seen a balance scale, right?
If there’s the same amount of weight on each side of the scale,
then the two sides are in balance.
But, if we add some weight to one side...
then the scale will tip.
The two sides are no longer in balance.
An equation is like that.
Whatever is on one side of the equal sign MUST have exactly the same value
as whatever is on the other side.
Otherwise, the equation would not be true.
Of course, that doesn’t mean that the two sides have to look the same.
For example, in the equation 1 + 1 = 2,
1 + 1 doesn’t LOOK the like the number 2,
but we know that 1 + 1 has the same VALUE as 2,
so 1 + 1 = 2 is in balance. It’s a true equation.
The reason we need to know that equations must be balanced
is because when we start rearranging them, if we are not careful,
we might do something that would change one of the sides more than the other.
That would make the equation get out of balance and it wouldn’t be true anymore.
And if that happens, we won’t get the right answer when we solve it.
That sounds pretty bad, huh? So how do we avoid that?
How do we avoid getting an equation out of balance?
The key is that whenever we make a change to an equation,
we have to make the exact same change on both sides
That’s so important, I’ll say it again.
Whenever we do something to an equation,
we have to do the same thing to BOTH sides.
For example, if we want to add something to one side of an equation,
we have to add that same thing to the other side.
And if we want to subtract something from one side of an equation,
then we have to subtract that same thing from the other side.
And it’s the same for multiplication and division.
If we want to multiply one side of an equation by a number,
then we need to multiply the other side by that same number.
Or if we want to divide one side of an equation by a number,
then we have to divide the other side by that number also.
As long as you always do the same thing to both sides of an equation,
it will stay in balance and your equation will still be true.
Alright, like I said, in this video, we’re just going to focus on equations involving addition and subtraction.
And here’s our first example: x + 7 = 15
To solve for the unknown value ‘x’,
we need to rearrange the equation so that the ‘x’ is all by itself on one side of the equal sign.
But what can we do to get ‘x’ all by itself?
Well, right now ‘x’ is not by itself because 7 is being added to it.
Is there a way for us to get rid of that 7?
Yes! Since seven is being added to the ‘x’, we can undo that by subtracting 7 from that side of the equation.
Subtracting 7 would leave ‘x’ all by itself because ‘x’ plus 7 minus 7 is just ‘x’.
The ‘plus 7’ and the ‘minus 7’ cancel each other out.
Okay great! So we just subtract 7 from this side of the equation and ‘x’ is all by itself.
…equation solved, right?
WRONG! If we just subtract 7 from one side of the equation and not the other side,
then our equation won’t be in balance anymore.
To keep our equation in balance, we also need to subtract 7 from the other side of the equation.
But on that side, we just have the number 15.
So we need to subtract 7 from that 15.
And since 15 - 7 = 8, that side of the equation will just become 8.
There, by subtracting 7 from BOTH sides, we’ve changed the original equation (x + 7 = 15)
into the new and much simpler equation (x = 8) which tells us that the unknown number is 8.
We have solved the equation!
And to check our answer, to make sure we got it right,
we can see what would happen if we replaced the unknown value in our original equation with the number 8.
Instead of x + 7 = 15, we’d right 8 + 7 = 15,
and if that’s true, then we know we got the right answer.
Let’s try another one: 40 = 25 + x
This time, the unknown value is on the right hand side of the equation.
Does that make it harder?
Nope. We use the exact same strategy.
We want to get ‘x’ by itself, but this time ‘x’ is being added to 25.
But thanks to the commutative property, that’s the same as 25 being added to ‘x’.
So, to isolate 'x', we should subtract 25 from that side of the equation.
But then we also need to subtract 25 from the other side to keep things in balance.
On the right side, x plus 25 minus 25 is just x
The minus 25 cancels out the positive 25 that was there.
And on the other side we have 40 minus 25 which would leave 15.
So the equation has become 15 = x, which is the same as x = 15.
Again, we’ve solved the equation.
So, whenever something is being added to an unknown,
we can undo that and get the unknown all by itself
by subtracting that same something from both sides of the equation.
But what about when something is being subtracted from an unknown,
like in this example: x - 5 = 16
In this case, ‘x’ is not by itself because 5 is being subtracter or taken away from it.
…any ideas about how we could get rid of (or undo) that ‘minus 5’?
Yep! To undo that subtraction, this time we need to ADD 5 to both sides of the equation.
The ‘minus 5’ and the ‘plus 5’ cancel each other out and leave ‘x’ all by itself on this side.
And on the other side, we have 16 + 5 which is 21.
So in this equation, x equals 21.
Let’s try another example like that: 10 = x - 32.
Again the ‘x’ is not by itself because 32 is being subtracted from it.
So to cancel that ‘minus 32’ out, we can just add 32 to both sides of the equation.
On the right side, the ‘minus 32’ and the ‘plus 32’ cancel out leaving just ‘x’.
And on the left we have 10 + 32 which is 42. Now we know that x = 42.
Okay, so now you know how to solve very simple equations like these
where something is being added to an unknown or where something is being subtracted from an unknown.
But before you try practicing on your own,
I want to show you a tricky variation of the subtraction problem
that confuses a lot of students.
Do you remember how subtraction does NOT have the commutative property?
If you switch the order of the subtraction, it’s a different problem.
Suppose we get a problem, where instead of a number being taken away from an unknown,
an unknown is being taken away from a number.
What do we do in that case?
Well, we still want to get the unknown all by itself,
but it’s a little harder to see how to do that.
In this problem (12 - x = 5) the 12 on this side is a positive 12,
so we could subtract 12 from both sides.
That would get rid of the 12,
but the problem is that wouldn’t get rid of this minus sign.
That’s because the minus sign really belongs to the ‘x’ since it’s the ‘x’ that is being subtracted.
Subtracting 12 would leave us with ‘negative x’ on this side of the equal sign,
which is not wrong, but it might be confusing if you don’t know how to work with negative numbers yet.
Fortunately, there’s another way to do this kind of problem that will avoid getting a negative unknown.
Instead of subtracting 12 from both sides, what would happen if we added ‘x’ to both sides?
Can we do that? Can we add an unknown to both sides?
We can add or subtract ANYTHING we want as long as we do it to both sides!
And when we do that, the ‘minus x’ and the ‘plus x’ will cancel each other out on this side.
And and the other side, we will get 5 + x.
Now our equation is 12 = 5 + x.
And you might be thinking, “but why would we do that? That didn’t even solve our equation!”
That’s true, but it changed it into an equation that we already know how to solve.
Now it’s easy to see that we can isolate the unknown just by subtracting 5 from both sides of the equation.
That will give us 7 = x or x = 7.
It just took us one extra step to rearrange the equation,
but then it was easy to solve.
Okay, that’s the basics of solving simple algebraic equations that involve addition and subtraction.
You just need to get the unknown value all by itself,
and you can do that by adding or subtracting something from both sides of the equation.
And this process works the same even if the numbers in the equations are decimals or fractions.
And it also works the same no matter what symbol you are using as an unknown.
It could be x, y, z or a, b, c.
The letter being used doesn’t matter.
Remember, when it comes to math,
it’s really important to practice what you’ve learned.
So be sure to try solving some basic equations on your own!
As always, thanks for watching Math Antics and I’ll see ya next time.