"Expressions" that represent the same value may appear in several different forms,
referred to as equivalent expressions.
equivPic

The Distributive Property ensures that 3x + 6 and 3(x + 2) are equivalent expressions.
To double check, we substituted the number 5 into each expression and got the result 21 from both.

You can show that expressions are equivalent:

• algebraically:   • numerically:
by showing, through algebraic computations, that both expressions can be represented as the same expression.
    - remove parentheses
    - combine similar terms
    - arrange terms from both
           expressions in the same order
    - keep working until both
           expressions are exactly the same
 
by showing, through numerical substitution, that the same number(s) replacing the variable(s) in both expressions yield the same numeric results.
    - substitute the same number(s) for each
           variable in each expression
    - compute the numerical results of each            expression
    - the numerical results will be the same for            both expressions
NOTE: Avoid choosing the number 0 for substituting.


hint gal
When determining equivalent expressions:
take your time and LOOK CAREFULLY!
Some expressions may not LOOK equivalent at first glance,
but upon further examination will be equivalent.

The number 3 will be used for the numerical checks in the following examples. 

expin1
Are these expressions equivalent?
    7x + 2x and 14x


Numerical check (x = 3)
7(3) + 2(3) = 27
14(3) = 42
27 ≠ 42
Algebraic check:
7x + 2x = 9x
9x ≠ 14x
Not Equivalent!
frown

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expin2
Are these expressions equivalent?
    5(x - 2) and 5x - 10


Numerical check (x = 3)
5(3 - 2) = 5
5(3) - 10 = 5
CHECK
Algebraic check:
5(x - 2) = 5x - 10
Distributive Property

CHECK
Equivalent!
smile

In this example, by the Distributive Property 5(x - 2) = 5(x) - 5(2) = 5x - 10.

dash

expin3
Are these expressions equivalent?
    6(3x) and 9x


Numerical check ( x = 3)
6(3(3)) = 54
9(3) = 27
54 ≠ 27
Algebraic check:
6(3x) = 18x
9x 18x
Not Equivalent!
frown

dash

expin4 Show, by completing the table, whether the given expressions are equivalent to 6(x + 2).
Use x = 3 for numerical checking.

The distributive property shows 6(x + 2) = 6x + 12. And when x = 3, 6(3 + 2) = 6(5) = 30

Expression
Y/N
Numerical Check
(Let x = 3)
Algebraic Check
a) 6x + 2
NO
6(3) + 2 = 20 30
6x + 2 6(x + 2) = 6x + 12

b) 3x + 6 + 3x + 6
YES
3(3) + 6 + 3(3) + 6 = 30
3x + 6 + 3x + 6 = 6x + 12

c) 6x + 12
YES
6(3) + 12 = 30
6x + 12 = 6(x + 2) = 6x + 12

d) 3(x + 2) + 3(x + 2)
YES
3(3 + 2) + 3(3 + 2) = 30
3(x + 2) + 3(x + 2) =
3x + 6 + 3x + 6 = 6x + 12

e) 3(x + 2) + 3x
NO
3(3 + 2) + 3(3) =
24 30
3(x + 2) + 3x = 3x + 6 + 3x
=
6x + 6 6x + 12

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expin5
Are these expressions equivalent?
    18x + 27 = 9(2x + 3)

Numerical check ( x = 3)
18(3) + 27 = 81
3(6(3)+9) = 81
CHECK
Algebraic check:
18x +27 = 9(2x +3)
18x + 27 = 18x + 27
Distributive Property applied
to the right hand side.
CHECK
Equivalent!
smile


In this example, we could also use the Distributive Property in reverse.
Start with 18x + 27.
Factor out the GCF of 9.
And we get 9(2x + 3). CHECK.

 

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