Combining Like Terms

TL;DR

Combining like terms means simplifying an algebraic expression by grouping and performing operations on terms that have the exact same variables and powers. Think of it like sorting different types of fruit; you can only add apples to apples. This process makes expressions much easier to understand and work with.

1. The Mental Model

Imagine you have a bunch of different items mixed up in a basket, like apples, bananas, and oranges. You can only combine the same types of fruit. In algebra, "like terms" are those identical fruits, and combining them means counting how many of each you have.

2. The Core Material

When you're simplifying algebraic expressions, combining like terms is a crucial step. It's all about making your expressions as short and clear as possible.

What are "Like Terms"?

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Like terms are terms that have:
1. The exact same variables. For example, 'x' and 'x' are the same, but 'x' and 'y' are different.
2. The exact same powers for those variables. For example, 'x²' and 'x²' are the same, but 'x²' and 'x³' are different. Also, 'xy' and 'yx' are like terms because the variables are the same, just in a different order (multiplication order doesn't matter).

Crucially, the coefficient (the number in front of the variable) doesn't matter when deciding if terms are 'like' or not. 3x and 5x are like terms because they both have 'x'.

Here’s a quick way to think about it:

graph TD
    A["Start with a Term"] --> B{("Does it have variables?")};
    B -- Yes --> C{("Are the variables the same as another term?")};
    B -- No --> D["Consider it a constant term"];
    C -- Yes --> E{("Are the powers of those variables the same?")};
    C -- No --> F["Not a like term (different variable)"];
    E -- Yes --> G["It's a Like Term!"];
    E -- No --> H["Not a like term (different power)"];
    G --> I["Combine its coefficients"];
    D --> I;

How to Combine Like Terms

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Once you identify like terms, combining them is simple:

1. Identify like terms: Look for terms that have identical variable parts (including their powers).
2. Group them (optional but helpful): You can physically rearrange the expression to put like terms next to each other. Remember to keep the sign (+ or -) that's in front of the term with it.
3. Add or subtract their coefficients: Once grouped, you just add or subtract the numbers in front of the variables. The variable part stays exactly the same.

Example 1: Simple variables
3x + 5x
Here, 3x and 5x are like terms because they both have 'x'.
Combine the coefficients: 3 + 5 = 8.
So, 3x + 5x = 8x.

Example 2: Variables with powers
4y² - 2y² + 7y
4y² and -2y² are like terms (both y²).
7y is not a like term with y² because the power of y is different (it's y or y¹).
Combine 4y² - 2y²: 4 - 2 = 2. So, 2y².
The 7y stays as it is.
Result: 2y² + 7y. (You can't combine these further).

Example 3: Constants
5a + 7 - 2a - 3
Like terms:
* 5a and -2a (both have 'a')
* +7 and -3 (both are constants - numbers without variables)

Group them: (5a - 2a) + (7 - 3)
Combine coefficients:
* 5a - 2a = 3a
* 7 - 3 = 4
Result: 3a + 4.

3. Worked Example

Let's simplify the expression: 7p + 3q - 2p + 5 - q - 8

1. Identify like terms:

   • Terms with p: 7p and -2p
   • Terms with q: 3q and -q (remember -q is the same as -1q)
   • Constant terms: +5 and -8

2. Group them together:
   (7p - 2p) + (3q - q) + (5 - 8)

3. Combine the coefficients for each group:

   • For p terms: 7 - 2 = 5. So, 5p.
   • For q terms: 3 - 1 = 2. So, 2q.
   • For constant terms: 5 - 8 = -3.

4. Write the simplified expression:
   5p + 2q - 3

4. Key Takeaways

• Like terms must have the exact same variables raised to the exact same powers.
• The order of variables in a term doesn't matter (e.g., xy is like yx).
• Only the coefficients of like terms are added or subtracted; the variable part stays untouched.
• Constant numbers are like terms with each other.
• Always keep the sign in front of a term when you move it or combine it.

Common Mistakes to Avoid:
- Combining terms with different variables: Don't combine 3x with 2y.
- Combining terms with different powers: Don't combine 4x² with 5x.
- Forgetting the sign: If you have -2y, the minus sign belongs to the 2y.
- Assuming no coefficient: If you see x, it means 1x, not 0x.

5. Now Try It

Simplify the following expression as much as possible: 10m + 4n - 3m - 7n + 2mn - 6

Your goal is to have an expression where no more terms can be combined, and you should have separate terms for m, n, mn, and any constant numbers. You'll know you're successful if your final expression is shorter and each term has a unique variable combination.