Just as we have rules in arithmetic, algebra also has laws that guide how we work with numbers and letters.
These rules are called the Laws of Algebra.
They help us simplify and rearrange expressions correctly.

1. Commutative Law

The commutative law means that the order of addition or multiplication does not affect the result.

• For addition:
  a + b = b + a
  Example: ( 3 + 5 = 5 + 3 = 8 )
• For multiplication:
  ab = ba
  Example: ( 2 × 4 = 4 × 2 = 8 )

❌ Note: The commutative law does not apply to subtraction or division.
That is, ( a - b ≠ b - a ) and ( a ÷ b ≠ b ÷ a ).

2. Associative Law

The associative law tells us that when adding or multiplying three or more numbers,
the way they are grouped does not change the answer.

• For addition:
  (a + b) + c = a + (b + c)
  Example: ( (2 + 3) + 4 = 2 + (3 + 4) = 9
• For multiplication:
  (ab)c = a(bc)
  Example: ( (2 × 3) × 4 = 2 × (3 × 4) = 24 )

3. Distributive Law

This law links multiplication and addition/subtraction together.

and

Example 1:
Simplify ( 3(x + 4) )
= 3x + 12

Example 2:
Simplify ( 2(a + b + c) )
= 2a + 2b + 2c

4. Identity Laws

These laws describe what happens when we add or multiply by 0 or 1.

• Additive Identity:
  ( a + 0 = a )
• Multiplicative Identity:
  ( a × 1 = a )

5. Inverse Laws

• Additive Inverse:
  ( a + (-a) = 0 )
• Multiplicative Inverse:
  ( a × \frac{1}{a} = 1 ) (for ( a ≠ 0 ))

Worked Examples

Example 1:
Simplify ( 5(x + 2) ).
= 5x + 10

Example 2:
Simplify ( 2(3a + 4b) + a(2 + b) ).
= 6a + 8b + 2a + ab = 8a + 8b + ab

Example 3:
Simplify ( (x + y) + (y + x) ).
Using commutative law:
x + y + y + x = 2x + 2y

Example 4:
Simplify ( 3(2a + b) - 2(a + b) ).
= 6a + 3b - 2a - 2b = 4a + b