Algebra is the language of mathematics, allowing us to explain connections and solve problems with symbols and variables. In the extensive field of algebra, adding and subtracting algebraic expressions is critical. These operations combine or decompose expressions, allowing us to simplify difficult equations, solve real-world issues, and lay a solid basis for advanced mathematical ideas.

## Understanding Algebraic Expressions:

An algebraic expression consists of constants, variables, and mathematical operations such as addition, subtraction, multiplication, and division. Examples of algebraic expressions are

- 3x + 5y
- 2x
- 2a/b + 3

In the first expression, '3x' and '5y' are terms, where '3' and '5' are constants, and 'x' and 'y' are variables. The second expression is a polynomial with terms '2x^{2}', '-7xy', and '4'. The third expression contains a fraction with '2a/b' as one of the terms.

### Addition of Algebraic Expressions

Addition of algebraic expressions involves combining like terms. Like terms are expressions that have the same variables which are raised to the same powers.

#### Let's take an example to illustrate this:

Example 1: Add the following algebraic expressions: 3x^{2} + 4xy - 2x^{2 }- 3xy + 7.

Solution: To add these expressions, we group the like terms together: (3x^{2} - 2x^{2} + (4xy - 3xy) + 7

Simplifying the like terms: X^{2} + xy + 7

Thus, the sum of the given expressions is 'x^{2} + xy + 7'.

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### Subtraction of Algebraic Expressions:

Subtraction of algebraic expressions follows the same principles as addition. We identify the like terms and then subtract the coefficients of those like terms while keeping the variables and their exponents unchanged.

Let's take another example:

Example 2: Subtract the following algebraic expressions: 5x^{2} - 3xy + 2 and 2x^{2} + 5xy - 3.

Solution: To subtract these expressions, we group the like terms together:

(5x^{2} - 2x^{2}) + (-3xy - 5xy) + (2 + 3)

Simplifying the like terms:

3x^{2} - 8xy + 5

Thus, the difference of the given expressions is '3x^{2} - 8xy + 5'.

Some more examples:

Example 1: Add the following algebraic expressions: 3a^{2} - 5ab + 2b^{2} and 2a^{2} + 3ab – b^{2}.

Solution: To add these expressions, we group the like terms together: (3a^{2} + 2a^{2}) + (- 5ab + 3ab) + (2b^{2} – b^{2}).

Simplifying the like terms: 5a^{2} - 2ab - b^{2}

Result: The sum of the given expressions is '5a^{2} - 2ab - b^{2 }'.

Example 2: Subtract the following algebraic expressions: 4x^{3} + 7x^{2} - 3x and x^{3} - 2x^{2} + 5x.

Solution: To subtract these expressions, we group the like terms together: (4x^{3} + x^{3 }) + (7x^{2} - 2x^{2}) + (- 3x + 5x) .

Simplifying the like terms: 3x^{3} + 5x^{2} + 8x

Result: The difference of the given expressions is '3x^{3} + 5x^{2} + 8x'

### Real-world applications of adding and subtracting algebraic expressions

The addition and subtraction of algebraic expressions have numerous applications in various fields, including physics, engineering, economics, and computer science.

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The addition and subtraction of algebraic expressions are basic functions that provide the foundation for addressing complicated problems in mathematics and its applications. Understanding these operations allows us to alter expressions, simplify equations, and obtain knowledge of diverse mathematical phenomena. As we continue on our mathematical adventure, keep in mind that understanding the fundamentals, such as adding and subtracting algebraic expressions, creates a solid basis for more complex ideas and improves our problem-solving abilities in a variety of fields inside mathematics and beyond.