Collecting Like Terms: Addition and Subtraction

TL;DR

Collecting like terms means combining pieces of an expression that are exactly the same type. You can only add or subtract terms that have identical variable parts, like adding apples to apples. Think of it as tidying up an algebraic expression to make it simpler and easier to work with.

1. The Mental Model

Imagine you're sorting toys: you put all the cars together, all the dolls together, and all the building blocks together. You don't try to add a car to a doll. Collecting like terms is exactly like that, but with numbers and letters.

2. The Core Material

When you're dealing with algebraic expressions, "terms" are the individual pieces separated by addition or subtraction signs. A "like term" is a term that has the exact same variable part, including the variable letters and their exponents. The number in front (the coefficient) can be different.

What are "Like Terms"?

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• 2x and 5x are like terms because they both have x.
• 3y² and -y² are like terms because they both have y².
• 7 and -10 are like terms because they are both just numbers (they don't have variables, so we call them constants).
• 4ab and ab are like terms because they both have ab.

What are "Unlike Terms"?

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• 2x and 5y are unlike terms (different variables).
• 3x and 3x² are unlike terms (different exponents on the variable).
• 4a and 4ab are unlike terms (different variable parts).

How to Collect Like Terms

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1. Identify the like terms in the expression. It helps to use different shapes or colors to group them visually.
2. Combine the coefficients (the numbers in front) of the like terms through addition or subtraction.
3. Keep the variable part exactly the same. Don't change the variables or their exponents.
4. Rewrite the simplified expression.

Here's how it generally flows:

graph TD
    A["Start with an expression"] --> B{"Identify Like Terms"};
    B --> C["Group terms with identical variable parts"];
    C --> D["Combine coefficients of each group"];
    D --> E["Keep the variable part unchanged"];
    E --> F["Write the simplified expression"];
    F --> G["End"];

Examples of Collecting Like Terms:

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Example 1: Simple Addition
3x + 5x
Both 3x and 5x are like terms because they both have x.
Combine the coefficients: 3 + 5 = 8.
Keep the variable: x.
Result: 8x

Example 2: Simple Subtraction
7y - 2y
Both 7y and 2y are like terms because they both have y.
Combine the coefficients: 7 - 2 = 5.
Keep the variable: y.
Result: 5y

Example 3: With Constants
4a + 6 + 2a - 3
Like terms with a: 4a and 2a.
Like terms (constants): 6 and -3.

Combine a terms: 4a + 2a = 6a.
Combine constants: 6 - 3 = 3.
Result: 6a + 3

Example 4: With Different Variables/Exponents
5x² + 2x - 3x² + 7x + 1
Like terms with x²: 5x² and -3x².
Like terms with x: 2x and 7x.
Constant term: 1.

Combine x² terms: 5x² - 3x² = 2x².
Combine x terms: 2x + 7x = 9x.
The constant 1 stays as is.
Result: 2x² + 9x + 1

3. Worked Example

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

1. Identify like terms:

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

2. Group them (mentally or by rewriting):
   (8p - 5p) + (3q + q) - 2

3. Combine coefficients for each group:

   • For p terms: 8 - 5 = 3. So, 3p.
   • For q terms: 3 + 1 = 4. So, 4q.
   • The constant -2 remains.

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

This is the most simplified form because 3p, 4q, and -2 are all unlike terms.

4. Key Takeaways

• You can only add or subtract terms that have the exact same variable part.
• When combining like terms, you only add or subtract their coefficients (the numbers in front).
• The variable part (letters and their exponents) stays exactly the same after combining.
• A term like x or y implicitly has a coefficient of 1 (e.g., x means 1x).
• Constants (numbers without variables) are like terms with other constants.

Common Mistakes

• Changing the variable part: Don't add exponents (e.g., x + x is 2x, not x²).
• Combining unlike terms: You can't combine 2x + 3y; they stay separate.
• Forgetting negative signs: Pay close attention to the sign in front of each term when combining.
• Treating x and x² as like terms: They are different because their exponents are different.

5. Now Try It

Simplify the following expression as much as possible: 7a² + 4b - 2a² - b + 5 + 3b

What to do: Identify the different types of terms, group them, and then add or subtract their coefficients.

What success looks like: You'll end up with a simplified expression containing no more than three terms, where each term has a unique variable part or is a constant.