Welcome to a lesson on equations and the story of X.
An algebraic equation is
a mathematical sentence stating that an algebraic expression is equal to
a specific value or variable or another expression. So an equation must have an
equal sign. And we've already had a lesson on expressions and the story of X.
This is a lesson that focuses on equations and the story of X. To describe the story of
X, we want to write out the steps that describe the operations being performed on X, or with X, as
well as the operations performed on the result of each step. So an example of one
we're asked to tell the story of X and the expression X plus seven, and use this to
determine the solution to the equation, X plus seven equals eighteen. So we know we can probably
solve this equation just by inspection. Let's go ahead and go through the story of X to solve this
equation. So under this first column, the story of X, we'll be able to give an equation by listing each step. So we
begin with X, because the expression is X plus seven on the left. In words we add seven,
which would give us the expression X plus seven. And because we have the expression X plus seven
equals eighteen, we want the result to be eighteen, which gives us the equation, X plus seven equals
eighteen. And now to solve, we'll start with the result, in this case eighteen, and undo each
operation to get back to X. So in this case we'll have to undo, add seven.
So we'll being with eighteen.
Now we want to undo or perform the opposite operation of add seven. So we subtract seven.
That'd give us eighteen minus seven which
equals eleven, which brings us back to X. So we know our solution is X equals eleven.
Now let's go ahead and check our solution, by rewriting the story of X, for the
given equation again to show this solution works. So we begin with
X, which we know is equal to eleven. Next we add seven.
So eleven plus seven is equal to eighteen, which is good news
because we want the result to be eighteen, which it is. Let's take a look at another
example. Here we're asked to tell the story of X, and the expression for X, and use this to determine the
solution to the equation for X equals twenty-eight. So first tell the story of X to build
the equation,
beginning with X.
Four X means four times X, which is equivalent to X times four, due to
the commutative property of multiplication. So for the next step of the story of X,
we multiply by four.
So that would be X times four, which equals four X. There's only one operation to be
performed on X because four X equals twenty-eight, we want the result to be twenty-eight. Which gives
us the equation, four X equals twenty-eight. And now we'll solve. So we start with the result
and then undo each
operation. In this case this one operation, to get back to X.
The opposite operation of multiplying by four is divide by four.
It should give us twenty-eight divided by four which equals seven, which brings us back to X. Our
solution is X equals seven. Let's go ahead and check our solution. So we have X, which
is equal to seven. So
going back to the story of X, we multiply by four.
Which would be seven times four or four times seven, which equals twenty-eight,
which is good because we want the result to be twenty-eight.
So
this verifies our solution, X equals seven is correct.
Next we're asked to tell the story of X for the expression, five X plus one, and use this to determine the solution to the
equation, five X plus one equals twenty-one. So we begin with X. Five X means five
times X, which is equivalent to X times five. So
the next step in this story is multiply by five,
so that'd be five X.
And because we have plus one, the next step is add one,
which gives us five X plus one.
We want the expression five X plus one to equal twenty-one. So we want the result to be twenty-one, which gives us
the equation, five X plus one equals twenty-one.
And now we'll solve. So we'll begin with twenty-one, and then we're going undo each operation in reverse order to get back
to X. So we want to undo, add one, so we subtract one.
This would be twenty-one minus one which equals twenty.
To undo multiplying by five, we divide by five.
So we have
twenty divided by five which equals four, which brings us back to X. Our solution is X equals four. Let's
go and check our solution. So we are going to rewrite the story of X. So we have X, which we know is equal
to four. We multiply by five, which would be four
times five,
which is equal to five times four, which equals twenty.
Next step in the story is add one. Twenty
plus one equals twenty-one. We want the result to be twenty-one,
which is what we have. Which verifies our
solution is correct.
We'll look at two more examples in part two. I hope you found this helpful.