Subtraction of Algebraic Expressions

Subtracting algebraic expressions involves adding the additive inverse of the second expression to the first. It requires changing the signs of the terms of the expression being subtracted and then combining the like terms. There are three components of an algebraic expression: 
 

Coefficients: These are constant values like -1, 4, etc. 

Variables: These are unknown quantities like x, y, z, etc. 

Operators: For subtraction, the operator is the minus (-) symbol.

When subtracting the algebraic expressions students should follow the list of rules:

1. Flip signs: Always flip the signs of each term of the second expression and perform addition. 
    
2. Combine like terms: Only like terms can be subtracted from one another, so group all like terms together.
    
3. Simplifying result: After all like terms are combined, there will be unlike terms remaining. Write the remaining unlike terms as they are, along with the like terms, to get the final result.

The following are the methods of subtraction of algebraic expressions:

• Horizontal Method
   
• Column Method

To apply the horizontal method for subtraction of algebraic expressions, use the following steps.

Step 1: Write both expressions in the same line using a minus sign in between.

Step 2: Remove the brackets and change the signs of the second expression.

Step 3: Combine the like terms.

Let’s apply these steps to an example:

Question: Subtract (2x-y+4) from (5x+3y-2)

Step 1: Write both expressions in the same line, (5x +3y-2)-(2x-y+4)

Step 2: Remove the brackets and change the signs of the second expression 5x+3y-2-2x+y-4

5x and 2x are like terms having the same variable x, similarly, 3y and -y are also like terms.

Step 3: Write like terms together: (5x-2x)+(3y+y)+(-2-4)

Answer: 3x+4y-6

When subtracting the algebraic expressions using the column method, we write the expressions one below the other. Make sure like terms are aligned in each column. Then change the signs of the second expression and add the expressions. 

For example, Subtract (2x-y+4) from (5x+3y-2)

Solution: Arrange the like terms vertically in columns
  5x   +  3y   -  2    ← Minuend (from which we subtract)
- 2x   -   y   +  4    ← Subtrahend (what we subtract)
-----------------------
  3x   + 4y   - 6

Therefore, upon subtracting (2x-y+4) from (5x+3y-2), we get 3x   + 4y   - 6

In algebra, subtraction has some characteristic properties. These properties are listed below:

• Subtraction is not commutative
  In subtraction, changing the order of the terms changes the result, i.e., A - B ≠ B - A

• Subtraction is not associative
  Unlike addition, we cannot regroup in subtraction. When three or more expressions are involved, changing the grouping changes the result.
  (A − B) − C ≠ A − (B − C)

• Subtraction is the addition of the opposite sign
  Subtracting an expression is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second term.
  A − B = A + (−B)

• Subtracting zero from an expression leaves the expression as is
  Subtracting zero from any expression results in the same algebraic expression: A - 0 = A

Tips and tricks are useful for students to efficiently deal with the subtraction of algebraic expressions. Some helpful tips are listed below:

• Always pay attention to signs before combining like terms.

• If two expressions have identical terms, cross them out before starting the subtraction. This makes the expressions shorter and provides more clarity due to fewer terms.

• Beginners and visual learners can benefit from using the box model or column method to avoid missing signs and mismatching terms.

• Algebraic expression: An algebraic expression is a combination of terms including variables, constants, and operators.

• Like terms: Terms having the same variables raised to the same power are like terms.

• Unlike terms: Terms having different variables or exponents, or both, are unlike terms.

• Coefficient: The number in a term that multiplies a variable in an algebraic expression is a coefficient.

• Simplification: When like terms are combined to reduce a longer expression, this process is known as simplification.