Math worksheet teaching how to combine like terms in algebraic expressions.

It looks like you've uploaded a worksheet titled "Combining Like Terms", which is designed to teach students how to simplify algebraic expressions by combining like terms. While the image contains instructions, examples, and common mistakes, it does not include a specific problem for me to solve — instead, it's a note-taking or instructional worksheet.

However, I can help you understand and complete this worksheet fully. Let’s go through it step by step and fill in the blanks, explain the concepts, and clarify the examples.

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✔ Title: Combining Like Terms

> One way to simplify an expression is to "combine like terms."

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🔹 What does it mean to combine like terms?

> Answer:
To combine like terms means to add or subtract the coefficients of terms that have the same variable(s) raised to the same power. Only like terms can be combined.

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🔹 Fill in the Blanks:

> You can only combine terms that have the same variable and the same exponent.

✔ Explanation:
Like terms must have identical variables with identical exponents. For example:
- $ 3x $ and $ 5x $ are like terms (same variable $ x $, same exponent 1).
- $ 2x^2 $ and $ 4x^2 $ are like terms.
- But $ 3x $ and $ 3x^2 $ are not like terms (different exponents).

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🔹 Examples

Let’s work through each example:

#### ✔ Example 1:
$$
2a + 3b - 4a
$$

Step 1: Use commutative property to group like terms:
$$
2a - 4a + 3b
$$

Step 2: Combine coefficients:
$$
(2 - 4)a + 3b = -2a + 3b
$$

✔ Final Answer: $ \boxed{-2a + 3b} $

> Note: The negative sign moves with the term it's attached to.

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#### ✔ Example 2:
$$
14m - 3n^2 - 2n + 3m
$$

Group like terms:
$$
14m + 3m - 3n^2 - 2n
$$

Combine:
$$
(14 + 3)m - 3n^2 - 2n = 17m - 3n^2 - 2n
$$

Now arrange in alphabetical order (and within letters, by exponent):

- $ m $ comes before $ n $
- $ n^2 $ comes before $ n $

So final answer:
$$
\boxed{17m - 3n^2 - 2n}
$$

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#### ✔ Example 3:
$$
5x + 4x - 6 + 5x^2
$$

Group like terms:
$$
5x + 4x + 5x^2 - 6
$$

Combine:
$$
(5 + 4)x + 5x^2 - 6 = 9x + 5x^2 - 6
$$

Now put in correct order: variables in alphabetical order, then from highest to lowest exponent.

- $ x^2 $ comes before $ x $
- Constant last

So:
$$
\boxed{5x^2 + 9x - 6}
$$

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🔹 Note: Ordering Your Answers

> All of your answers should be arranged so that the variables are in alphabetical order first, then in order from greatest to least exponent.

✔ Example: $ 3y + 2x^2 - 4x $ → becomes $ 2x^2 - 4x + 3y $

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🔹 Common Mistakes – Identify & Correct

#### ✘ Mistake #1:
$$
a^2 - 4a + 5a \quad \text{→ incorrectly simplified to } 2a^2
$$

🔴 Error: You cannot combine $ a^2 $ and $ a $ because they are not like terms.

✔ Correct simplification:
$$
a^2 + (-4a + 5a) = a^2 + a
$$

✔ Final Answer: $ \boxed{a^2 + a} $

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#### ✘ Mistake #2:
$$
3y + 4x^2 - 3y + 5y
$$

Incorrectly simplified as: $ 7y + 4x^2 $

🔴 Error: Variables not in correct order (should be alphabetical), and terms were combined incorrectly.

Let’s fix it:

Group like terms:
- $ 3y - 3y + 5y = (3 - 3 + 5)y = 5y $
- $ 4x^2 $ stays

So: $ 4x^2 + 5y $

✔ Now in correct order: $ \boxed{4x^2 + 5y} $

> Always list variables alphabetically: $ x $ before $ y $

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#### ✘ Mistake #3:
$$
3h + 14g - 5h + 5g
$$

Incorrectly simplified as: $ 9g + 8h $

But the steps show:
- $ 3h - 5h = -2h $, not $ 8h $
- $ 14g + 5g = 19g $, not $ 9g $

🔴 Error: Signs were not moved with the terms correctly.

Correct grouping:
- $ 3h - 5h = -2h $
- $ 14g + 5g = 19g $

Now arrange: $ g $ before $ h $ → $ 19g - 2h $

✔ Final Answer: $ \boxed{19g - 2h} $

> Also, negative signs must move with the term: $ -5h $, not just $ -5 $

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✔ Summary of Rules:

1. Only combine like terms (same variable and same exponent).
2. Use commutative property to group like terms.
3. Move negative signs with the term they're attached to.
4. Arrange answers in alphabetical order of variables, then by descending exponents.
5. Never combine unlike terms (e.g., $ x $ and $ x^2 $).

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📝 Final Notes for Students:

- Practice identifying like terms first.
- Watch out for signs — especially when subtracting!
- Always check your order: alphabetical, then highest to lowest exponent.

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If you have a specific problem you'd like me to solve using these rules, feel free to type it in! Otherwise, this worksheet is now fully explained and corrected.