Addition and Subtraction of Algebraic Expressions

In this section, we will learn how to add and subtract algebraic expressions, a fundamental operation in algebra. Algebraic expressions are mathematical phrases involving variables, constants, and arithmetic operations. The process of addition and subtraction requires combining like terms. Let's break it down step by step.

Like Terms

In algebra, like terms are terms that have the same variable(s) raised to the same power(s). For example, $3x$ and $5x$ are like terms because both contain the variable $x$. On the other hand, $3x$ and $4y$ are not like terms because they have different variables.

Adding Algebraic Expressions

To add algebraic expressions, follow these steps:

1. Identify the like terms in both expressions.
2. Add the coefficients of like terms while keeping the variable part the same.
3. Rewrite the result by combining the simplified terms.

For example:

Add the following expressions: $3x + 5y$ and $4x + 2y$.

Step 1: Identify the like terms:

• Like terms: $3x$ and $4x$ (both have the variable $x$)
• Like terms: $5y$ and $2y$ (both have the variable $y$)

Step 2: Add the coefficients of like terms:

• For $x$: $3x + 4x = 7x$
• For $y$: $5y + 2y = 7y$

Step 3: Combine the results:

The simplified expression is: $7x + 7y$

Subtracting Algebraic Expressions

To subtract algebraic expressions, follow these steps:

1. Identify the like terms in both expressions.
2. Subtract the coefficients of like terms while keeping the variable part the same.
3. Rewrite the result by combining the simplified terms.

For example:

Subtract the following expressions: $7x + 3y$ and $4x + 5y$.

Step 1: Identify the like terms:

• Like terms: $7x$ and $4x$ (both have the variable $x$)
• Like terms: $3y$ and $5y$ (both have the variable $y$)

Step 2: Subtract the coefficients of like terms:

• For $x$: $7x - 4x = 3x$
• For $y$: $3y - 5y = -2y$

Step 3: Combine the results:

The simplified expression is: $3x - 2y$

Important Points to Remember

• Only like terms can be added or subtracted. Terms with different variables or different powers cannot be combined.
• When subtracting, remember to distribute the negative sign across the terms of the second expression.
• Ensure you maintain the proper signs when adding or subtracting the coefficients.

Let's look at one more example:

Example 1: Add the expressions $2a + 3b$ and $4a - 2b$.

Step 1: Identify like terms:

• Like terms: $2a$ and $4a$ (both have the variable $a$)
• Like terms: $3b$ and $-2b$ (both have the variable $b$)

Step 2: Add the coefficients:

• For $a$: $2a + 4a = 6a$
• For $b$: $3b - 2b = b$

Step 3: Combine the results:

The simplified expression is: $6a + b$

Example 2: Subtract the expressions $5x - 4y$ and $2x + 3y$.

Step 1: Identify like terms:

• Like terms: $5x$ and $2x$ (both have the variable $x$)
• Like terms: $-4y$ and $3y$ (both have the variable $y$)

Step 2: Subtract the coefficients:

• For $x$: $5x - 2x = 3x$
• For $y$: $-4y - 3y = -7y$

Step 3: Combine the results:

The simplified expression is: $3x - 7y$