"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

dash

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

dash

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.

 

divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use".

Topical Outline | JrMath Outline | MathBitsNotebook.com | MathBits' Teacher Resources
Terms of Use
   Contact Person: Donna Roberts

Copyright © 2012-2025 MathBitsNotebook.com. All Rights Reserved.