Addition of Algebraic Expressions: Definition, Components, Method, Rules, and Solved Examples

The addition of algebraic expressions is an important concept in algebra that involves combining like terms to simplify and solve expressions. An algebraic expression includes variables, constants, and operators, and adding these expressions requires identifying and summing the coefficients of like terms – terms with the same variables and exponents. Understanding this topic is important for solving equations and is frequently tested in school exams, and standardized tests like the SAT, SSC, Railways, and other math assessments. Keep reading to understand this addition of algebraic expression principles through solved examples.

What is Algebraic Expression?

An algebraic expression is a mathematical phrase that combines variables, constants, and operators (such as addition, subtraction, multiplication, and division) to represent a value that can change based on the variables’ values. Key components of an algebraic expression include:

• Variables: Symbols (e.g., x, y) representing unknown values.
• Constants: Fixed numbers (e.g., 5, -3) that do not change.
• Coefficients: Numerical factors multiplying the variables (e.g., 3 in 3x).
• Terms: Individual parts of the expression separated by addition or subtraction signs (e.g., in 2x+3y−5, the terms are 2x, 3y, and −5).
• Like Terms: Terms with the same variables raised to the same power (e.g., 4x and −2x are like terms).
• Unlike terms: Terms that have different variables or different exponents.
• Degree of a term: The sum of the exponents of the variables in a term.
• Degree of a polynomial: The highest degree of any term in the polynomial.

Important properties related to algebraic expressions include:

• Commutative Property: Addition and multiplication are commutative, meaning a+b=b+a and ab=ba.

a+b = b+a and ab = ba

• Associative Property: The way terms are grouped in addition or multiplication does not change their sum or product, respectively. For example, (a+b)+c=a+(b+c).

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

• Distributive Property: Multiplication distributes over addition or subtraction, as in a(b+c)=ab+ac.

a(b+c) = ab+ac

• Identity Property: Adding 0 to a number or multiplying a number by 1 doesn’t change the value. Example: a + 0 = a and a x 1 = a

a + 0 = a and a x 1 = a

• Inverse Property: Adding the opposite of a number to itself results in 0. Multiplying a number by its reciprocal results in 1. Example: a + (-a) = 0 and a x (1/a) = 1

a + (-a) = 0 and a x (1/a) = 1

Also Read: Define Line in Maths: 9 Types and Examples

Components of Algebraic Expressions

The components of algebraic expressions are fundamental elements that work together to form a complete mathematical statement. Here’s a breakdown of these components:

Components	Description	Examples
Variable	Symbols that represent unknown values.   Commonly represented by letters like x, y, z, a, b, etc.	In the expression 3x + 5, x is the variable.
Constants	Fixed numerical values.   They don’t change.	In the expression 3x + 5, 5 is the constant.
Coefficients	Numbers that are multiplied by variables.   They tell us how many times the variable is present.	In the expression 3x + 5, 3 is the coefficient of x
Terms	Individual components of an expression separated by addition or subtraction.   Each term can be a variable, a constant, or a product of variables and constants.	In the expression 3x + 5, 3x and 5 are the terms.
Operations	Arithmetic operations like addition, subtraction, multiplication, and division.   They combine the other components of the expression.	In the expression 3x + 5, + is the operation.

Example Breakdown

Let’s break down the expression 4xy – 2z + 7:

• Variables: x, y, z
• Constants: 7
• Coefficients: 4, -2
• Terms: 4xy, -2z, 7
• Operations: -, +

Methods to Solve Addition of Algebraic Expressions

Adding algebraic expressions involves combining like terms. There are primarily two methods to do this:

1. Horizontal Method

• Step 1: Write all the expressions in a horizontal line separated by plus signs, enclosing each expression in parentheses.
• Step 2: Remove the parentheses and group like terms together.
• Step 3: Add the coefficients of like terms and retain the common variable.

Example: Add: 3x + 2y, 5x – 3y, and x + 4y

• (3x + 2y) + (5x – 3y) + (x + 4y)
• 3x + 2y + 5x – 3y + x + 4y
• (3x + 5x + x) + (2y – 3y + 4y)
• 9x + 3y

2. Vertical Method

• Step 1: Write the expressions one below the other, aligning like terms in columns.
• Step 2: Add the coefficients of like terms column-wise.

Example: Add: 3x + 2y, 5x – 3y, and x + 4y

  (3x + 2y) + (5x – 3y) + (x + 4y) = 9x + 3y

Important points to remember:

• Only like terms can be added.
• The variables and their exponents remain unchanged during addition.
• The result is a simplified algebraic expression.

Rules of Addition of Algebraic Expressions

When adding algebraic expressions, follow these important rules of addition of Algebraic Expression:

1. Identify Like Terms:

• Like terms have the same variables raised to the same powers.
• Example: 3x, -5x, and 2x are like terms.

2. Combine Like Terms:

• Add the coefficients of like terms. The variable part remains unchanged.
• Example: 3x + (-5x) + 2x = (3 – 5 + 2)x = 0x = 0

3. Arrange Terms:

• It’s often helpful to arrange terms in descending order of their exponents for better organization.

4. Simplify:

• Combine the results of step 2 to get the final simplified expression.

Example: Add: 4x² + 3xy – 2y² and -2x² + 5xy + y²

• Identify like terms:
  • 4x² and -2x²
  • 3xy and 5xy
  • – 2y² and y²
• Combine like terms:
  • 4x²  + (-2x²) = 2x²
  • 3xy + 5xy = 8xy
  • – 2y² + y² = -y²
• Arrange terms:
  • 2x² + 8xy – y²

Remember:

• Only like terms can be added.
• The variables and their exponents do not change during addition.
• The result is a simplified algebraic expression.

Addition of Algebraic Expressions Solved Examples

Here are five solved examples demonstrating the addition of algebraic expressions:

Example 1: 3x + 4x

• Combine Like Terms: Both terms are like terms (both have x).
• Add Coefficients: 3x + 4x = (3+4)x = 7x
• Solution: 7x

Example 2: 5a + 2b + 3a − b

• Combine Like Terms: Group like terms: 5a and 3a are like terms, and 2b and −b are like terms.
• Add Coefficients: (5a+3a)+(2b−b) = 8a+b
• Solution: 8a+b

Example 3: 2(x+3) + 4(x−1)

• Distribute: Apply the distributive property. 2(x+3)=2x+6 

4(x−1) = 4x−4

• Combine Like Terms: Add the results. 2x+6+4x−4 = (2x+4x)+(6−4) = 6x+2
• Solution: 6x + 2

Example 4: 7m−2(n+4)+3m

• Distribute: Apply the distributive property to the term with parentheses. 

−2(n+4)=−2n−8

• Combine Like Terms: Add the results. 7m+3m−2n−8 = (7m+3m)−2n−8 = 10m−2n−8
• Solution: 10m−2n−8

Example 5: (4x²+2x−3) + (3x²−5x+7)

• Combine Like Terms: Group like terms. (4x²+3x²) + (2x−5x) + (−3+7)
• Solution: 7x²−3x+4

Also Read: Rational And Irrational Numbers: Differences, Examples

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