Answer key for a combining like terms math worksheet with simplified expressions.

It looks like you've uploaded an answer key for a math worksheet titled "Combining Like Terms", which focuses on simplifying algebraic expressions using distribution, addition, subtraction, and multiplication. The worksheet includes 15 problems where students are asked to simplify expressions by distributing and combining like terms.

Let me walk through the process of solving one of these problems step-by-step (using the first one as an example), then explain how all the answers were derived.

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🔍 Example Problem:

Simplify: $ 2(2x + 3) $

#### Step 1: Use the Distributive Property
Distribute the 2 to both terms inside the parentheses:

$$
2 \cdot 2x = 4x \\
2 \cdot 3 = 6
$$

So:
$$
2(2x + 3) = 4x + 6
$$

✔ This matches the answer given in the key.

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Now let’s go over how each problem was solved using the same method:

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✔ Problem-by-Problem Explanation

| Expression | Steps | Simplified |
|----------|-------|-----------|
| 1. $ 2(2x + 3) $ | $ 2 \cdot 2x = 4x $, $ 2 \cdot 3 = 6 $ → $ 4x + 6 $ | $ 4x + 6 $ ✔ |
| 2. $ 3(4 - 3y) $ | $ 3 \cdot 4 = 12 $, $ 3 \cdot (-3y) = -9y $ → $ -9y + 12 $ | $ -9y + 12 $ ✔ |
| 3. $ -2(5a + 7) $ | $ -2 \cdot 5a = -10a $, $ -2 \cdot 7 = -14 $ → $ -10a - 14 $ | $ -10a - 14 $ ✔ |
| 4. $ -3(n + 8) $ | $ -3 \cdot n = -3n $, $ -3 \cdot 8 = -24 $ → $ -3n - 24 $ | $ -3n - 24 $ ✔ |
| 5. $ 5(-2m + 7) $ | $ 5 \cdot (-2m) = -10m $, $ 5 \cdot 7 = 35 $ → $ -10m + 35 $ | $ -10m + 35 $ ✔ |
| 6. $ 4(3 - 9y) $ | $ 4 \cdot 3 = 12 $, $ 4 \cdot (-9y) = -36y $ → $ -36y + 12 $ | $ -36y + 12 $ ✔ |
| 7. $ 7(5b - 4) $ | $ 7 \cdot 5b = 35b $, $ 7 \cdot (-4) = -28 $ → $ 35b - 28 $ | $ 35b - 28 $ ✔ |
| 8. $ 2(9 + 5a) $ | $ 2 \cdot 9 = 18 $, $ 2 \cdot 5a = 10a $ → $ 10a + 18 $ | $ 10a + 18 $ ✔ |
| 9. $ -6(-4x - 4) $ | $ -6 \cdot (-4x) = 24x $, $ -6 \cdot (-4) = 24 $ → $ 24x + 24 $ | $ 24x + 24 $ ✔ |
| 10. $ 5(-7r + 6) $ | $ 5 \cdot (-7r) = -35r $, $ 5 \cdot 6 = 30 $ → $ -35r + 30 $ | $ -35r + 30 $ ✔ |
| 11. $ 8(8 - 3k) $ | $ 8 \cdot 8 = 64 $, $ 8 \cdot (-3k) = -24k $ → $ -24k + 64 $ | $ -24k + 64 $ ✔ |
| 12. $ 9(6b - 2) $ | $ 9 \cdot 6b = 54b $, $ 9 \cdot (-2) = -18 $ → $ 54b - 18 $ | $ 54b - 18 $ ✔ |
| 13. $ 4(-4c - 9) $ | $ 4 \cdot (-4c) = -16c $, $ 4 \cdot (-9) = -36 $ → $ -16c - 36 $ | $ -16c - 36 $ ✔ |
| 14. $ -2(10 - 3z) $ | $ -2 \cdot 10 = -20 $, $ -2 \cdot (-3z) = +6z $ → $ 6z - 20 $ | $ 6z - 20 $ ✔ |
| 15. $ 3(-9m + 4) $ | $ 3 \cdot (-9m) = -27m $, $ 3 \cdot 4 = 12 $ → $ -27m + 12 $ | $ -27m + 12 $ ✔ |

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🧠 Key Concepts Used:

1. Distributive Property:
$ a(b + c) = ab + ac $

2. Sign Rules:
- Negative times positive = negative
- Negative times negative = positive

3. Like Terms:
Only combine terms with the same variable and exponent (e.g., $ 4x $ and $ 6x $), but here we're just distributing — no combining needed after distribution since each expression becomes two separate terms.

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✔ Final Notes:

All answers on the answer key are correct and follow the distributive property accurately.

If this was a student worksheet, they would be expected to:
- Apply the distributive property.
- Multiply coefficients correctly.
- Pay attention to signs (especially negatives).
- Write the final expression in standard form (variable term first, then constant).

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Let me know if you'd like a printable version, a blank worksheet, or help explaining any specific problem!