Hello there, budding mathematician! 👋 It's fantastic that you're digging into the nuances of combining like terms in Algebra 1. It's truly one of the foundational skills that makes algebraic expressions manageable and helps solve equations. Think of me as your friendly guide through this essential concept!
What Exactly Are "Like Terms"? 🤔
Before we combine them, we need to know what we're looking for! Like terms are terms that have the exact same variable(s) raised to the exact same power(s). The numerical part, called the coefficient, can be different – that's what we combine!
- Example 1: $\mathbf{3x}$ and $\mathbf{7x}$ are like terms because they both have the variable $\mathbf{x}$ raised to the first power.
- Example 2: $\mathbf{5y^2}$ and $\mathbf{-2y^2}$ are like terms because they both have $\mathbf{y}$ raised to the second power.
- Example 3: $\mathbf{4ab}$ and $\mathbf{10ab}$ are like terms.
- NOT Like Terms: $\mathbf{3x}$ and $\mathbf{3x^2}$ are NOT like terms (different powers). Also, $\mathbf{5x}$ and $\mathbf{5y}$ are NOT like terms (different variables). And a constant like $\mathbf{8}$ is a like term only with other constants.
The Golden Rule: Why Can We Combine Them? 🤝
The ability to combine like terms stems directly from the distributive property! When you have something like $\mathbf{3x + 5x}$, it's essentially saying "3 groups of x plus 5 groups of x." Using the distributive property, we can write this as $\mathbf{(3+5)x = 8x}$. It’s just like saying "3 apples + 5 apples = 8 apples." You can't combine apples and oranges!
How to Combine Like Terms: Step-by-Step! 📝
Let's break down the process into easy-to-follow steps:
- Identify Like Terms: Scan your expression and group the terms that are "alike." It often helps to use different colors, underline them, or circle them on paper. Remember to include the sign ($\mathbf{+}$ or $\mathbf{-}$) in front of each term!
- Combine the Coefficients: Once you've identified your like terms, simply add or subtract their numerical coefficients. This is where your integer rules come in handy!
- Keep the Variable Part the Same: The variable(s) and their exponent(s) do NOT change when you combine like terms. They are simply carried along.
- Rewrite the Simplified Expression: Put all your combined terms back together, usually in alphabetical order for variables and with constants at the end.
Examples in Action! ✨
Let's try a few:
- Example A: Simplify $\mathbf{4x + 7y - x + 2y}$
Like terms for $\mathbf{x}$: $\mathbf{4x}$ and $\mathbf{-x}$ (which is $\mathbf{-1x}$)
Like terms for $\mathbf{y}$: $\mathbf{7y}$ and $\mathbf{2y}$
Combine $\mathbf{x}$ terms: $\mathbf{4x - x = 3x}$
Combine $\mathbf{y}$ terms: $\mathbf{7y + 2y = 9y}$
Result: $\mathbf{3x + 9y}$
- Example B: Simplify $\mathbf{5a^2 + 3a - 2a^2 + 8}$
Like terms for $\mathbf{a^2}$: $\mathbf{5a^2}$ and $\mathbf{-2a^2}$
Like terms for $\mathbf{a}$: $\mathbf{3a}$ (no other $\mathbf{a}$ terms)
Like terms for constants: $\mathbf{8}$ (no other constants)
Combine $\mathbf{a^2}$ terms: $\mathbf{5a^2 - 2a^2 = 3a^2}$
Result: $\mathbf{3a^2 + 3a + 8}$
Pro Tip! 💡 Always remember that a variable by itself, like $\mathbf{x}$ or $\mathbf{y}$, has an invisible coefficient of $\mathbf{1}$. So, $\mathbf{x}$ is really $\mathbf{1x}$, and $\mathbf{-y}$ is $\mathbf{-1y}$. This helps immensely when combining!
You've got this! Practice is key to making this second nature. Keep an eye on those signs and variable powers, and you'll be simplifying expressions like a pro in no time! Happy math-ing! 🚀