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Combining Like Terms and Distributive Property in Algebra
Combining like terms and the distributive property are fundamental concepts in algebra. Mastering these skills is essential for simplifying algebraic expressions and solving equations. Let's break down these concepts and explore how they work together.
What are Like Terms?
Like terms are terms in an algebraic expression that have the same variables raised to the same powers. For example:
- 3x and 5x are like terms because they both have the variable 'x' raised to the power of 1.
- 2y2 and -7y2 are like terms because they both have the variable 'y' raised to the power of 2.
- 4 and 9 are like terms because they are both constants (numbers without variables).
Combining Like Terms
To combine like terms, simply add or subtract their coefficients (the numbers in front of the variables). Remember that the variable and its exponent stay the same.
Example:
Simplify the expression: 3x + 5x - 2y + 7y
Solution:
Combine the 'x' terms: 3x + 5x = 8x
Combine the 'y' terms: -2y + 7y = 5y
The simplified expression is: 8x + 5y
The Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each term of the sum by that number.
Formula: a(b + c) = ab + ac
Example:
Simplify the expression: 2(x + 3)
Solution:
Distribute the 2: 2 * x + 2 * 3
Simplify: 2x + 6
Combining Like Terms and the Distributive Property
Often, you'll need to use both the distributive property and combining like terms to simplify an expression. Here's an example:
Simplify the expression: 3(2x - 1) + 4x + 5
Solution:
1. Distribute the 3: 6x - 3 + 4x + 5
2. Combine the 'x' terms: 6x + 4x = 10x
3. Combine the constant terms: -3 + 5 = 2
The simplified expression is: 10x + 2
Practice Problems
Here are some practice problems to solidify your understanding:
- Simplify: 7a - 2a + 5b - 3b
- Simplify: 4(x + 2) - 3x
- Simplify: 2(3y - 1) + 5(y + 2)
Remember, practice is key to mastering any math concept. By understanding how to combine like terms and apply the distributive property, you'll be well on your way to success in algebra!