It’s Professor Dave; let’s solve some equations.
We now understand what algebraic equations are, and in order to solve increasingly complicated
equations, we will have to continually develop new mathematical tools.
So let’s learn how to solve algebraic equations, starting with some very basic ones.
Take something like x plus two equals five.
What this is telling us is that there is some number x, such that when we add two to it,
we get five.
We can probably just use common sense to realize that this number must be three, because when
we add three and two, we get five.
But this kind of reasoning won’t work with trickier equations, so we have to learn a
more rigorous approach to solve for a variable.
In order to solve for a variable, we have to look at any operations that variable is
participating in, and systematically undo each of those operations by doing something
to both sides of the equation.
We can do anything we want to this equation as long as we do it to both sides, because
if both sides are equal, and we do the same thing to both sides, the sides will remain
equal.
So let’s look at this equation, what is happening to x?
Well, two is being added to x.
How do we undo this?
We simply perform the inverse operation, and the inverse of addition is subtraction.
We have to subtract two from both sides.
On the left, x plus two minus two leaves x all by itself, which was precisely the plan.
On the right, we get five minus two, which is three, and therefore X equals three, the
answer we were expecting.
If we had something like N minus four equals seventeen, we would add four to both sides,
because addition is the inverse of subtraction.
On the left we get N by itself, on the right we get twenty-one, so N is equal to twenty-one.
This works the same way for multiplication and division.
Given the equation 4D equals 20, we can see that the variable is being multiplied by four.
The inverse of multiplication is division, so we have to divide both sides by four.
On the left, the fours cancel, and on the right, twenty divided by four is five, so
D equals five.
And if we have Y over eight equals three, we now have to multiply both sides by eight,
because the inverse of division is multiplication.
On the left, the eights cancel, leaving us with Y, and on the right, three times eight
is twenty-four, so Y equals twenty-four.
How about something a little harder, like four minus B equals seven.
The order might look a little confusing, but we still just need to get rid of this four,
so let’s subtract four from both sides.
That will leave negative B equals seven minus four, which is three.
We want to know the value of positive B, so we just have to either multiply or divide
both sides by negative one.
That will mean that B equals negative three.
With this example we can see that if we have two steps to do, we undo operations in the
reverse order of PEMDAS.
We work with addition and subtraction first, and then we do multiplication or division.
This is the order that is necessary to get the variable by itself.
Algebraic equations can certainly get much more complicated than this, but we should
master these simple ones first, so let’s check comprehension.