SIMPLIFYING ALGEBRAIC EXPRESSIONS

A term is a number, a variable, or a product or quotient of numbers and variables. For example 5, x, 7a, b², and 2m³n are all terms. Like terms contain identical variables. For example, in 5x² - 3x² + 3x, the terms 5x² and -3x² are like terms because the variable part of each term is identical.

The coefficient of a term is a number that multiplies a variable. For example, in 10x², the coefficient of x² is 10, and in 4m/5, the coefficient m is 4/5.

An expression is in simplest form when it is replaced by an equivalent expression having no like terms or parentheses. Simplifying means rewriting in simpler form.

Distributive Property

For any real numbers a, b and c,

a(b + c) = ab + ac

a(b - c) = ab - ac

Examples :

5(2 + 7) = 5 ⋅ 2 + 5 ⋅ 7

5(7 - 3) = 4 ⋅ 7 - 4 ⋅ 3

Commutative Property

For any real numbers a, b and c,

a + b = b + a

a ⋅ b = b ⋅ a

Examples :

5 + 8 = 8 + 5

5 ⋅ 8 = 8 ⋅ 5

Associative Property

For any real numbers a, b and c,

(a + b) + c = a + (b + c)

(ab)c = a(bc)

Examples :

(2 + 3) + 4 = 2 + (3 + 4)

(2 ⋅ 3) ⋅ 4 = 2 ⋅ (3 ⋅ 4)

Solved Problems

Simplify the following algebraic expressions :

Problem 1 :

4x³ - 2(x³ + 3x)

Solution :

= 4x³ - 2(x³ + 3x)

Using Distributive Property,

= 4x³ - 2x³ - 2x

Combining like terms,

= 2x³ - 2x

Problem 2 :

5.4(x - 2y) - 2.7(x - 3y)

Solution :

= 5.4(x - 2y) - 2.7(x - 3y)

Using Distributive Property,

= 5.4x - 10.8y - 2.7x + 8.1y

Combining like terms,

= 2.7x - 2.7y

Problem 3 :

(1/2)(2a + 3b + 4c) - (3/2)(b + 2c)

Solution :

= (1/2)(2a + 3b + 4c) - (3/2)(b + 2c)

Using Distributive Property,

= a + 3b/2 + 2c - 3b/2 - 3c

Combining like terms,

= a - c

Problem 4 :

a(b - c) - b(a + c) - c(a - b)

Solution :

= a(b - c) - b(a + c) - c(a - b)

Using Distributive Property,

= ab - ac - ab - bc - ac + bc

Combining like terms,

-2ac

Problem 5 :

3[6a - 3(1 - a) - 5(a + 1)]

Solution :

=  3[6a - 3(1 - a) - 5(a + 1)]

Using Distributive Property,

=  3[6a - 3 + 3a - 5a - 5]

Combining like terms,

=  3[4a - 8]

Using Distributive Property,

=  12a - 24

Problem 6 :

(2/3)(y² - y - 3) + (1/3)(y² + 2y + 6)

Solution :

=  (2/3)(y² - y - 3) + (1/3)(y² + 2y + 6)

Using Distributive Property,

=  2y²/3 - 2y/3 - 6/3 + y²/3 + 2y/3 + 6/3

=  2y²/3 + y²/3 - 2y/3 + 2y/3 - 6/3 + 6/3

Combining like terms,

=  (2y² + y²)/3

=  3y²/3

=  y²

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