Addition and Subtraction of Algebraic Expressions

In Mathematics, we mainly have four basic operations: addition, subtraction, multiplication, and division. These can be performed on numbers and algebraic expressions. The addition and subtraction of algebraic expressions is almost similar to the addition and subtraction of numbers. However, in the case of algebraic expressions, we need to sort and place the like terms and the unlike terms together which makes it easier to simplify. 

• We can only combine like terms by adding or subtracting them with one another.
• Unlike terms cannot be combined by adding or subtracting.

The terms whose variables and exponents are the same are known as 'like terms' and the terms having different variables are 'unlike terms'. Let us understand this with an example.

• Observe this expression: 6x² + 7x + 8x² + 9 + 8y
• Notice the variables of each term. There are two terms with the same variable (x²): 6x² and 8x²
• That means 6x² and 8x² like terms and thus can be added or subtracted from each other.
• When adding or subtracting like terms, the coefficients are added (or subtracted) and the variable remains unchanged. So, the above expression can be simplified to: 14x² + 7x + 9 + 8y

Addition of Algebraic Expressions

Horizontal method 

Let us solve these expressions with the help of the horizontal method: (p + 2q + 3r + 4) + (2p + 4q + 6r + 2). To do this, we will first combine the like terms, write them together and add them to get the answer.

• Step 1: Open the brackets: p + 2q + 3r + 4 + 2p + 4q + 6r + 2
• Step 2: Combine the like terms to get the simplified expression: 3p + 6q + 9r + 6.

Column method 

The column method requires the expressions to be written column-wise, one below the other, taking care that like terms are to be placed in the same column. Then, we add the numerical coefficient of each column (like terms) and write the sum below it followed by the common variable. Let's add the same expressions using the column method.

Subtraction of Algebraic Expressions

While subtracting one algebraic expression from another, we need to be careful with the signs. It should be noted that if there’s a subtraction sign before the brackets, we need to reverse all the signs once we open the brackets.

Horizontal Method:

Let's solve: (6a + 2b - 3c) - (4a - 4b + 9c + 12) using the horizontal method.

• Step 1: Open the brackets: 6a + 2b - 3c - 4a + 4b - 9c -12 (Observe the changed signs of the second expression)
• Step 2: Combine the like terms to get the simplified expression: 2a + 6b - 12c - 12.

Column Method:

Let's subtract the same expressions using the column method. As we place the two expressions one below the other, we change all the signs of the second number as shown below and then we simplify the expressions as per their signs.

Tips and Tricks 

• While adding expressions, remember that “(3x + 5y)” is not equal to “(8xy)”.
• A useful technique when simplifying expressions is to use different patterns of underlining for different sets of like terms. This makes it easy to add and subtract.

Important Notes

• Addition and Subtraction of algebraic expressions can be done by two methods: horizontal method and column method.
• It is always better to subtract two expressions at one time. Do not try to subtract three or more expressions together through the column method.
• In case there is a negative sign outside the brackets, operators inside the brackets need to be changed.

Topics related to Addition and Subtraction of Algebraic Expressions

• Addition of Algebraic Expressions
• Subtraction of Algebraic Expressions
• Multiplication of Algebraic Expressions
• Division of Algebraic Expressions

What Is an Algebraic Expression?

An algebraic expression (or) a variable expression is made up of variables and constants combined by operations such as addition, subtraction, multiplication, division, etc.

Can you Add or Subtract Unlike Terms?

No, for addition or subtraction of algebraic expressions we consider only the like terms. The like terms are placed together and then simplified. Unlike terms cannot be added or subtracted.

What Are the Rules in Multiplying Algebraic Expressions?

The rules in multiplying algebraic expression are as follows:

• The product of two factors with the same signs will be positive.

• The outcome of multiplying two terms with two unlike signs will be negative.

How Do you Simplify Expressions and Equations?

Given below are the basic steps to be followed to simplify an algebraic expression:

• Remove the brackets, if any.
• Use exponent rules to remove parentheses for the terms with exponents.
• Combine like terms and add or subtract them by adding or subtracting coefficients.
• Combine the constants and add or subtract them.

What are the Basics of Algebra?

Algebra is a branch of Mathematics that uses letters to find out the numbers that are not known. The letters are called variables and the values which are known are called constants. The basics of Algebra involve simple mathematical operations: addition, subtraction, multiplication, and division, operating on both constant as well as variables. For example, y + 3 = 0 is an equation where 'y' is the variable and 3 is the constant. Here, the value of 'y' can be found by solving the equation, which will result in 'y' = -3.