In this activity,
we are given an expression,
asked to simplify the expression,
and then evaluate both expressions
for a given value of the variable.
Then determine if the expressions are equivalent.
In the first row,
we are given the expression
Five w minus two, times the quantity of four minus two w.
Because we have minus two here,
what is equivalent to plus negative two
to clear the parentheses,
we distribute negative two.
So, beginning with the given expression,
we distribute negative two
which gives us five w
and then negative two times four
is equal to negative eight.
Instead of writing plus negative eight,
let's write minus eight,
which is equivalent.
And then we have negative two
times negative two w, which is positive four w,
which gives us plus four w.
The next step is to combine like terms.
We have two w terms.
Five w and four w are like terms.
Five w plus four w is nine w.
The expressions simplifies to nine w minus eight.
And now we want to evaluate both expressions,
meaning the original and the simplified expression,
when w equals negative two.
Let's first evaluate the original expression
by substituting negative two for w here
and here.
Which will give us five times negative two
minus two times the quantity four
minus two times negative two.
Following the order of operations,
we simplify within the innermost parentheses first.
Inside these parentheses here, we have subtraction
and multiplication.
We multiply first.
Two times negative two is equal to negative four.
This simplifies to four minus negative four,
or four plus four.
So let's write this as five times negative two
minus two times four plus four.
Again, we could have written this as four
minus negative four,
which is equivalent to plus four.
Still simplifying inside the parentheses,
We need to find the sum of four and four,
which is eight.
This simplifies to five times negative two
minus two times eight.
Next step is to multiply.
Five times negative two is equal to negative ten.
And then we have minus two times eight,
and therefore, we have minus sixteen
which equals negative twenty-six.
And now, we want to evaluate
the simplified expression when w equals negative two,
which gives us nine times negative two,
minus eight.
Multiplying before subtracting.
Nine times negative two is equal to negative eighteen.
Negative eighteen minus eight
is equal to a negative twenty-six.
Whenever we evaluate an expression
and the simplified form of the expression
for a particular value of the variable,
we should always get the same value.
And that's because both expressions are equivalent.
So we enter yes in this last column
asking if the expressions are equivalent.
For the second example,
there are no parentheses to clear.
And therefore, the first step is to combine like terms.
Starting with the original expression,
three x y and four x y are like terms.
Negative six x and negative two x
are also like terms.
Three x y plus four x y is seven x y.
And negative six x minus two x is negative eight x.
And instead of writing plus negative eight x,
it's more common to write minus eight x,
which is equivalent.
And now we want to evaluate both expressions
when x equals three and y equals negative one.
Beginning with the original expression,
performing the substitution three x y
is three times three times negative one.
Minus six x
Is minus six times three
plus four x y is plus four times three times negative one
and minus two x is minus two times three.
To simplify here, we begin by multiplying.
Three times three times negative one is negative nine.
And negative nine minus six times three is eighteen.
And then we have plus four times three times negative one
is negative twelve.
And then we have minus two time three,
which is minus six.
Now, adding and subtracting from left to right,
negative nine minus eighteen is negative twenty-seven.
Negative twenty-seven plus negative twelve
is negative thirty-nine.
Negative thirty-nine minus six is equal
to negative forty-five.
Now, we want to evaluate the simplified expression.
Substituting three for x and negative one for y,
gives us seven time three times negative one
minus eight time three.
Multiplying before subtracting,
seven times three times negative one
is negative twenty one minus eight times three
is twenty-four.
Negative twenty-one minus twenty-four is equal to
negative forty-five.
Just as expected, we get the same value using
either expression.
And that is because both expressions are equivalent.
I hope you found this helpful.