The Distributive Property can be applied to simplify expressions. For example, the expression 2 × (128 + 12) can be solved using two different methods.
Method 1
2 × (128 + 12) = 2(140) First, add.
= 280 Then,
multiply.
Method 2
2 × (128 + 12) = (2 × 128) + (2 × 12) First,
distribute.
= 256 + 24 Multiply.
= 280 Add.
The Distributive Property is used in Method 2. Using both
methods, the
value of the expression is 280.
|
Distributive Property |
Symbols: For
any numbers a, b, and c, a(b + c) = ab + ac and a(b – c) = ab – ac. 2(5 – 3) = (2 ⋅ 5) – (2 ⋅ 3) |
In the expression a(b + c), it does not matter
whether a is placed to the left or to the right of the expression in
parentheses. So, (b – c)a = ba – ca and (b – c)a
= ba – ca.
Example
Simplify each expression.
1. 3(x + 7)
Alternative Solutions :
3(x + 7) = (3 ⋅ x) + (3 ⋅ 7) Distributive
Property
= 3x
+ 21 Substitution Property
2. 5(2n + 8)
Alternative Solutions :
5(2n + 8) = (5 ⋅ 2n) + (5 ⋅ 8) Distributive
Property
= 10n + 40 Substitution
Property
A term
is a number, variable, or product or quotient of numbers and variables.
|
Examples
of Terms |
Not
Terms |
||
|
7 |
7 is a number |
7 + x |
7 + x is
the sum of two terms |
|
t |
t is a variable |
8rs + 7y + 6 |
8rs + 7y + 6 is the sum of three terms. |
|
5x |
5x is a
product |
x – y |
x – y is the difference of two
terms |
The numerical part of a term that contains a variable is
called the coefficient.
For example, the coefficient of 2a is 2. Like terms are terms that contain the same
variables, such as 2a and 5a or 7xy and 3xy.
Consider the expression 5b + 3b + x + 12x.
•
There
are four terms.
•
The
like terms are 5b and 3b, x and 12x.
•
The
coefficients are shown in the table.
|
Term |
Coefficient |
|
5b 3b x 12x |
5 3 1 12 |
The Distributive Property allows us to combine like terms.
If a(b + c) = ab + ac,
then ab + ac a(b + c) by the Symmetric Property of
Equality.
2n + 7n = (2
+ 7)n Distributive Property
= 9n Substitution Property
The expressions 2n + 7n and 9n are
called equivalent
expressions because their values are the same for any value of n.
An algebraic expression is in simplest form when it has no like terms and no parentheses.
Example
Sports Link
3. Write an equation representing the area A of a soccer field given its width w and length l as shown in the diagram. Then simplify the expression and find the area if w is 54 yards and l is 60 yards.
Alternative Solutions :
Method
1
A = w(l + l) Multiply
the total length by the width.
= 54(60 + 60) Replace w with 54 and l
with 60.
= 54(120) Substitution
Property
= 6480 Substitution
Property
Method
2
A = wl + wl Add the areas of the smaller rectangles.
= 54(60) + 54(60) Replace w with 54 and l
with 60.
= 3240 + 3240 Substitution Property
= 6480 Substitution
Property
Using
either method, the area of the soccer field is 6480 square yards.
Sumber
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