In this activity,
we are given an expression,
we are asked to simplify the expression,
and then evaluate both expressions
for a given value of a variable or variables,
then answer the question:
Are the expressions equivalent?
In this first row,
we have three times the quantity x minus two plus seven x.
To simplify this expression,
the first step is to distribute the three.
Distributing the three we have three times x,
which is three x, and then three times negative two,
which is negative six, or minus six,
and then plus seven x.
The next step is to combine like terms.
We have two x terms. Three x plus seven x is equal to 10x.
The simplified expression is 10x minus six.
In the next column,
we're asked to evaluate both expressions when x equals six.
We will first substitute six for x
in the original expression,
which will give us three times the quantity six minus two,
plus seven times six.
Following the order of operations,
we simplify inside the parentheses first.
This simplifies to three times six minus two is four,
plus seven times six.
The next step is to multiply.
Three times four is 12, 12 plus seven times six is 42.
12 plus 42 is equal to 54.
Now we'll substitute six for x in the simplified expression,
which is 10x minus six.
Performing the substitution, give us 10 times six minus six.
Multiplying before subtracting, we have 60 minus six
which also equals 54.
As long as we simplify correctly,
we should get the same value for either expression
given a particular value of the variable.
The answer to, are the expressions equivalent,
the answer is always yes,
again assuming we simplified correctly.
The second example, there are no parentheses to clear,
therefore to simplify, we combine like terms.
Beginning with the given expression,
it might be helpful to write minus x as minus one x.
Combining like terms: negative three y squared
and two y squared are like terms.
So are four x and negative one x.
Simplifying, negative three y squared, plus two y squared
is negative one y squared or just negative y squared.
Then four x, minus one x is equal to three x,
we have plus three x.
Now we evaluate both expressions,
when x equals six and y equals negative four.
First, in the original expression,
we have negative three times y squared
is the square of negative four,
and then plus four times x is four times six
plus two times y squared
is two times the square of negative four
and then minus x is minus six.
The next step is to simplify the exponents.
The square of negative four is equal to 16.
This gives us negative three times 16,
plus four times six, plus two times 16, minus six.
Next step is to multiply.
Negative three times 16 is negative 48,
plus four times six is 24, plus two times 16 is 32,
then we still have minus six.
Adding and subtracting from left to right,
negative 48 plus 24 is negative 24.
Negative 24 plus 32 is equal to eight,
and eight minus six is equal to two.
The original expression is equal to two
when x equals six and y equals negative four.
Now we'll perform the substitution
into the simplified expression,
which would give us the opposite of y squared
is the opposite of the square of negative four,
plus three times x is three times six.
Simplifying the exponents first,
the square of negative four is 16,
so we have negative 16, plus 18,
which is also equal to positive two.
Which again is what we expect,
because these two expressions are always equivalent
as long as we simplify correctly.
So the answer here is yes.
I think we'll stop here
and look at two more examples in the next video.