In this video,
we will compare the deconstruct method
and the traditional method
for solving a linear equation
in one variable.
While the methods are different
it's important to recognize
that for both methods
we do perform the same operations
to both sides of the
equation, which results
in an equivalent equation.
For review, for the deconstruct method
also referred to as the
story of the variable method,
step one is to construct
the story of the variable.
Step two: deconstruct
the story of the variable
and then step three: apply
the deconstruct story
to both sides of the equation
to solve the equation.
Step four: check the solution.
However, some issues can
arise when using this method.
If an equation is in
the form of, let's say,
this equation here
where we have five
minus two x equals nine,
before we apply this method,
we need to write an equivalent equation
with the variable term first
as negative two x plus five equals nine.
There's also an issue when we
have variables on both sides
before we apply this method,
you would need to add
an equivalent equation
with the variable term on
one side of the equation
as shown here.
For these reasons it may
be helpful to transition
to a more traditional approach
to solving linear equation.
So, let's go through these steps as well.
Using the traditional method,
if the equation has decimals or fractions,
one option is to multiply
both sides of the equation
to clear the fractions or decimals
and then step one ...
But normally the first step,
because this step is optional,
the first step is to simplify
each side of the equation
by clearing parenthesis
and combining like terms.
Then add or subtract to
isolate the variable term
on one side and then finally,
multiply or divide to
isolate the variable and
solve the equation.
And again, check the solution.
So, going back to our example,
let's first construct
the story of the variable
that creates this equation
and then we'll write
the deconstruct story.
Starting with the variable z,
to construct the equation,
we first multiply by two
to get two z.
Then add five to the product.
Then, because you have
a negative three, here,
the next step is to
multiply by negative three
and then finally add three to the product
and the result is 30.
For the deconstruct story,
we want to undo the steps
of the construction story
which means we first undo the plus three
by subtracting three,
and then we undo multiplying
by negative three
by dividing by negative three.
The next step is to undo
adding five by subtracting five
and then finally we
undo multiplying by two
by dividing by two.
Let's begin by subtracting
three on both sides
of the equation.
Simplifying both sides, three
plus three is zero on the left
and 30 minus three is 27 on the right.
The equation simplifies
to negative three times
the quantity two z plus five
equals 27.
The next step is to divide
both sides by negative three.
Simplifying: negative three
divided by negative three
is one.
The equation simplifies to two z plus five
equals, 27 divided by negative
three is negative nine.
Next step is to undo the
plus five by subtracting five
on both sides.
Simplifying:
five minus five is zero.
We now have the equation two z equals
negative nine minus five is negative 14.
Last step is to undo multiplying
by two by dividing by two.
Simplifying: two divided by two is one.
One times z is z.
On the right, negative 14
divided by two is negative seven.
Our solution is z equals negative seven.
And now to solve this equation again
using a more traditional method.
Again, because you don't have
any fractions or decimals
the first step is to simplify
both sides of the equation
which means, in this case,
we distribute negative three
to begin.
So, negative three times
two z is negative six z.
Negative three times five is negative 15
plus negative 15 is equivalent to minus 15
and we still have plus
three equals thirty.
We can still simplify the
left side of the equation
by combining like terms.
Negative 15 plus three
is equal to negative 12.
We can rewrite the left side as
negative six z plus
negative 12 or minus 12
equals 30.
The next step is to add
or subtract to isolate
the variable term.
The variable term is negative six z.
We want to undo the minus 12.
In order to undo minus 12,
we add 12 to both sides
of the equation.
Simplifying.
Negative 12 plus 12 is zero.
We now have the equation
negative six z equals 42.
The last step is to multiply
or divide to solve for z.
Negative six z means negative six times z
and therefore to undo the multiplication
we divide both sides by negative six.
Negative six divided
by negative six is one.
One times z is z.
On the right,
42 divided by negative
six is negative seven.
So, of course, we do get the same solution
using the two different methods.
But again it's important to
remember for both methods
we did perform the same
operation to both sides
of the equation each time,
creating an equivalent equation.
Let's verify the solution is correct
by substituting negative
seven for z in the equation
to make sure it satisfies the equation.
Performing the substitution, we have,
negative three times the
quantity two times negative seven
plus five, plus three equals thirty.
Simplifying the left side of the equation,
we simplify inside the parenthesis first.
We multiply before adding.
This simplifies to negative three times
the quantity negative 14 plus five
plus three equals 30.
Still simplifying inside the parenthesis,
negative 14 plus five is negative nine.
We have negative three times
negative nine plus three
equals 30.
Simplifying on the left.
The next step is to
multiply: negative three
times negative nine is 27.
27 plus three equals 30.
27 plus three is 30.
30 equals 30 is true,
verifying our solution is correct.
I hope you found this helpful.
