1. Introduction
Adding and subtracting algebraic expressions is very similar to working with numbers. The only difference is that in algebra, we combine like terms. You have already learned how to identify like and unlike terms. Now we use that knowledge to simplify expressions by addition and subtraction.
The key idea is simple: only like terms can be added or subtracted. Unlike terms remain as they are.
2. Basic Rule: Combine Like Terms
The most important rule in addition and subtraction of algebraic expressions is:
Combine only the terms that have the same variable with the same power.
Example:
- \(5x + 3x = 8x\)
- \(7y - 2y = 5y\)
- \(4a + 2b\) cannot be combined
2.1. Why Only Like Terms Can Be Combined
Like terms represent the same type of quantity. Adding them is just like adding apples with apples, but not apples with oranges.
For example, \(3x + 2x\) means three groups of \(x\) plus two more groups of \(x\).
3. Horizontal and Vertical Methods
Addition and subtraction of expressions can be done using two methods: the horizontal method and the vertical method. Both give the same result.
3.1. Horizontal Method
In this method, write all expressions in a single line and then combine like terms.
Example:
\((3x + 5) + (7x - 2) = 3x + 7x + 5 - 2 = 10x + 3\)
3.2. Vertical Method
Write expressions one below the other and align like terms in the same column.
Example:
| \(3x + 5\) |
| \(+\ 7x - 2\) |
| \(10x + 3\) |
4. Addition of Expressions
To add expressions, arrange like terms together and then add their coefficients.
Example:
\((4x + 3y) + (2x - y) = 4x + 2x + 3y - y = 6x + 2y\)
4.1. Worked Examples
Example 1:
\((5a + 2b) + (3a - b) = 5a + 3a + 2b - b = 8a + b\)
Example 2:
\((7m - 4n + 2) + (3m + n - 5) = 7m + 3m - 4n + n + 2 - 5 = 10m - 3n - 3\)
5. Subtraction of Expressions
While subtracting expressions, remember to change the signs of the terms in the expression being subtracted.
Example:
\((6x + 4y) - (2x - y) = 6x - 2x + 4y + y = 4x + 5y\)
5.1. Worked Examples
Example 1:
\((8a - 3b) - (2a + b) = 8a - 2a - 3b - b = 6a - 4b\)
Example 2:
\((10m + 7) - (4m - 5) = 10m - 4m + 7 + 5 = 6m + 12\)
6. Common Mistakes
- Forgetting to change signs during subtraction.
- Trying to combine unlike terms.
- Ignoring negative signs in coefficients.
- Misaligning terms in the vertical method.
7. Mixed Examples with Step-by-Step Identification
Let’s simplify each expression:
7.1. Example Set
1. \((3x + 2y - 5) + (7x - y + 3)\)
Combine like terms:
- \(3x + 7x = 10x\)
- \(2y - y = y\)
- \(-5 + 3 = -2\)
Answer: \(10x + y - 2\)
2. \((9a - 4b + 7) - (3a + b - 2)\)
Change signs:
\(9a - 4b + 7 - 3a - b + 2\)
- \(9a - 3a = 6a\)
- \(-4b - b = -5b\)
- \(7 + 2 = 9\)
Answer: \(6a - 5b + 9\)
8. Quick Practice
Simplify the following:
- \((5x + 3) + (2x - 4)\)
- \((7a - 2b) - (3a - b)\)
- \((4m + n) + (6m - 3n)\)
- \((9p - 5) - (p + 2)\)
9. Summary
- Add and subtract only like terms.
- In subtraction, change the signs of the expression being subtracted.
- Use horizontal or vertical method for clarity.
- Always keep track of signs while combining terms.