Introduction to Algebraic Expressions:
Algebraic expressions are mathematical expressions that contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. These expressions are used to represent real-life situations and solve various mathematical problems.

Algebraic Expressions and Identities:
Algebraic expressions can be simplified and manipulated using algebraic identities. These identities are mathematical equations that hold true for all values of the variables involved. They are used to simplify complex expressions and solve equations.

Basic Algebraic Identities:
1. Identity for Addition: The sum of any number and zero is equal to the number itself.
Example: a + 0 = a

2. Identity for Multiplication: The product of any number and one is equal to the number itself.
Example: a × 1 = a

3. Commutative Property of Addition: The order of addition does not affect the result.
Example: a + b = b + a

4. Commutative Property of Multiplication: The order of multiplication does not affect the result.
Example: a × b = b × a

5. Associative Property of Addition: The grouping of numbers does not affect the result.
Example: (a + b) + c = a + (b + c)

6. Associative Property of Multiplication: The grouping of numbers does not affect the result.
Example: (a × b) × c = a × (b × c)

7. Distributive Property: Multiplication distributes over addition.
Example: a × (b + c) = (a × b) + (a × c)

Examples:
Let's consider a few examples to understand algebraic expressions and identities better.

Example 1: Simplify the expression 3x + 2y - x + 5y.
Solution: Combining like terms, we get 2x + 7y.

Example 2: Prove the identity (a + b)^2 = a^2 + 2ab + b^2.
Solution: Expanding the left-hand side, we get (a + b)(a + b) = a^2 + 2ab + b^2. Hence, the identity is proved.

Example 3: Solve the equation 2(x + 3) = 10.
Solution: Expanding and simplifying, we get 2x + 6 = 10. Solving for x, we find x = 2.

Algebraic expressions and identities play a crucial role in solving various mathematical problems. Understanding and applying these concepts can simplify complex expressions, solve equations, and make mathematical calculations easier.