About Distributive Property of Algebraic Expressions
What is the distributive property of algebraic expressions?
The distributive property of algebraic expressions indicates that we need to multiply each term in either the sum or the difference in an expression by a value outside the parentheses. The value outside the parentheses with the sum or difference is a number.
For example, (3x + 4y) multiplied by 4x, or (5y + 2) multiplied by 3, are examples of the distributive property when applied to algebraic expressions.
As you have understood, an algebraic expression is a combination of variables, operations, and constants.
For Sum
The distributive property for multiplication when the sum of two values is considered, is (a + b).c = ac + bc
Let us consider three numbers, 1, 5, and 2, to verify the above property.
So, (1 + 5).2 = 1. 2 + 5. 2
= 2 + 10
= 12
For Difference
The distributive property for multiplication when the difference of two values is considered, is (a – b).c = ac – bc
Let us consider three numbers, 1, 4, and 2, to verify the above property.
So, ( 4 – 1 ). 2 = 4. 2 – 1. 2
= 8 – 2
= 6
How do we solve problems based on applying the distributive property?
The solution is given below:
Step 1: Increase the number of variables in the equation.
Step 2: Multiply (distribute) the first, the outer, the inner, and the last numbers in each set.
Step 3: Combine similar terms.
Step 4: Solve the equation and, if necessary, simplify it.
Solved Distributive Property Algebraic Expressions Examples
Example 1: Simplify the given algebraic expressions using the distributive property.
(a) \(-\frac{1}{4}(8a-16)\)
(b) 5 ( – 2a + 4b )
(c) – 2 ( 1 + 4a – 5)
(d) \((4a+2)-2((\frac{3}{8})a-2)\)
Solution:
Part (a):
\(-\frac{1}{4}(8a-16)\)
\(=-\frac{1}{4}(8a)-[-\frac{1}{4}(16)]\) Using the distributive property
= – 2a – [ – 4 ] Multiply the terms
= – 2a + 4 Two opposite negative signs multiplied together become a positive sign
Part (b):
5 ( – 2a + 4b )
= 5 ( – 2a ) + 5 ( 4b ) Using the distributive property
= – 10a + 20b Multiply the terms
Part (c):
We can solve the above expression in two different ways:
1: Before combining the like terms, use the distributive property.
– 2 ( 1 + 4a – 5)
= – 2 ( 1 ) + [ -2 ( 4a ) ] – [ -2 ( 5 ) ] Using the distributive property
= -2 – 8a + 10 Multiply the terms
= -8a + 8 Combine the like terms
2: Before using the distributive property, add like terms in parentheses.
Addition of like terms
2 ( 1 + 4a – 5)
= – 2 ( 4a – 4)
= -2 ( 4a ) – 2 ( -4 ) Using the distributive property
= -8a + 8 Multiply the terms
Part (d):
\((4a+2)-2((\frac{3}{8})a-2)\)
\(=(4a+2)-2((\frac{3}{8})a)-2(-2)\) Using the distributive property
\(=(4a+2)-(\frac{3}{4})a+4\) Multiply the terms
\(=4a-(\frac{3}{4})a+2+4\) Group the like terms
\(=\frac{13a}{4}+6\)
Example 2: A square hall has a side length of m feet. To add tiles on the border of the hall, calculate the number of 3-foot square tiles that are required?
Solution:
Draw a diagram to describe the details in the question. With the help of the diagram, write an expression for the number of tiles required.
If we see the diagram above then the horizontal border should be taken m + 6 and the vertical border should be taken as as m. The tiled border can be divided into two sections, each requiring m + 3 + 3 tiles, and two sections, each requiring m tiles, as shown in the diagram.
As a result, the equation 2(m + 6) + 2m can be used to represent the number of tiles.
Simplify the given equation.
= 2 ( m + 6 ) + 2m
= 2 ( m ) + 2 ( 6 ) + 2m Using the distributive property
= 2m + 12 + 2m Simplify
= 4m + 12
Hence, the expression 4m + 12 will determine the number of 3-foot tiles required.
The distributive property instructs us on how to solve the expressions that can be written in the general form as
“a( b + c )”.The distributive property is also known as the multiplication and division distributive law.
You can use the distributive property regardless of the order of the factors because multiplication is commutative.
The properties of distribution state that for any real numbers a, b, and c, we use the following formulas:
Addition distributed by multiplication: ab + ac = a(b + c). Multiplication is distributive over subtraction: ab – ac = a(b – c)
The parentheses move in the associative Law, but the numbers or letters do not. When we add or multiply, the associative law comes into play. When we subtract or divide, this does not work. The law of distribution states, “multiply everything inside parentheses by what is outside it.”