Comprehensive algebra worksheet featuring ten problems focused on simplifying linear expressions using distribution and combining like terms.

It looks like you've uploaded a worksheet titled "Simplifying Linear Expressions" with 10 problems involving simplifying algebraic expressions. Since I can't view images directly, I'll help you solve each problem step by step based on the text you provided.

Let me go through each problem and simplify the expressions:

---

1. $ 3(2x - 4) + 5(x + 2) $

Step-by-step:
- Distribute:
$$
3(2x) - 3(4) + 5(x) + 5(2) = 6x - 12 + 5x + 10
$$
- Combine like terms:
$$
(6x + 5x) + (-12 + 10) = 11x - 2
$$

✔ Answer: $ \boxed{11x - 2} $

---

2. $ 2(8x + 7) - 3(2x + 1) + 4(2x + 5) $

Step-by-step:
- Distribute:
$$
2(8x) + 2(7) - 3(2x) - 3(1) + 4(2x) + 4(5) = 16x + 14 - 6x - 3 + 8x + 20
$$
- Combine like terms:
$$
(16x - 6x + 8x) + (14 - 3 + 20) = 18x + 31
$$

✔ Answer: $ \boxed{18x + 31} $

---

3. $ 5(2x + 6) + 9(2 - 4x) + 15 $

Step-by-step:
- Distribute:
$$
5(2x) + 5(6) + 9(2) - 9(4x) + 15 = 10x + 30 + 18 - 36x + 15
$$
- Combine like terms:
$$
(10x - 36x) + (30 + 18 + 15) = -26x + 63
$$

✔ Answer: $ \boxed{-26x + 63} $

---

4. $ 9(3x + 4) + 7(6 - 3x) + 3(4 - 3x) $

Step-by-step:
- Distribute:
$$
9(3x) + 9(4) + 7(6) - 7(3x) + 3(4) - 3(3x) = 27x + 36 + 42 - 21x + 12 - 9x
$$
- Combine like terms:
$$
(27x - 21x - 9x) + (36 + 42 + 12) = -3x + 90
$$

✔ Answer: $ \boxed{-3x + 90} $

---

5. $ -4(6x + 2) - 3(-2 + 4x) + 2(7x + 5) $

Step-by-step:
- Distribute:
$$
-4(6x) - 4(2) -3(-2) -3(4x) + 2(7x) + 2(5) = -24x - 8 + 6 - 12x + 14x + 10
$$
- Combine like terms:
$$
(-24x - 12x + 14x) + (-8 + 6 + 10) = -22x + 8
$$

✔ Answer: $ \boxed{-22x + 8} $

---

6. $ \frac{1}{2}(4x - 3) - \frac{2}{3}(x + \frac{1}{2}) + 2(7x) $

Step-by-step:
- Distribute:
$$
\frac{1}{2}(4x) - \frac{1}{2}(3) - \frac{2}{3}(x) - \frac{2}{3}(\frac{1}{2}) + 2(7x)
$$
$$
= 2x - \frac{3}{2} - \frac{2}{3}x - \frac{1}{3} + 14x
$$
- Combine like terms:
- Combine $ x $-terms:
$$
2x - \frac{2}{3}x + 14x = (16x - \frac{2}{3}x) = \frac{48}{3}x - \frac{2}{3}x = \frac{46}{3}x
$$
- Constant terms:
$$
-\frac{3}{2} - \frac{1}{3} = -\left(\frac{9}{6} + \frac{2}{6}\right) = -\frac{11}{6}
$$
- Final expression:
$$
\frac{46}{3}x - \frac{11}{6}
$$

✔ Answer: $ \boxed{\frac{46}{3}x - \frac{11}{6}} $

---

7. $ 4(2x + 6)(3x - 1) + 5(2x + 3) $

Wait — this one has two variables in parentheses, so it's not linear unless we expand it.

But let’s check: $ (2x + 6)(3x - 1) $ is a product of two binomials → will result in a quadratic term.

So this expression is not linear. But maybe it was meant to be simplified as-is?

Let’s assume it's a typo or misprint — but if it's correct, then we must expand:

Step-by-step:

First, expand $ (2x + 6)(3x - 1) $:
$$
= 2x(3x) + 2x(-1) + 6(3x) + 6(-1) = 6x^2 - 2x + 18x - 6 = 6x^2 + 16x - 6
$$

Now multiply by 4:
$$
4(6x^2 + 16x - 6) = 24x^2 + 64x - 24
$$

Then add $ 5(2x + 3) = 10x + 15 $

Total:
$$
24x^2 + 64x - 24 + 10x + 15 = 24x^2 + 74x - 9
$$

⚠️ This is a quadratic expression, not linear.

But since the worksheet says "Simplifying Linear Expressions", this might be an error.

Possibility: Maybe it's supposed to be $ 4(2x + 6) + 5(2x + 3) $? Let's try that.

Assume typo: $ 4(2x + 6) + 5(2x + 3) $

Then:
$$
= 8x + 24 + 10x + 15 = 18x + 39
$$

But based on what's written: $ 4(2x + 6)(3x - 1) + 5(2x + 3) $, it's quadratic.

If the problem is correct, answer is: $ \boxed{24x^2 + 74x - 9} $

But since the title says "linear", likely a typo.

Let me know if you want to proceed assuming it's linear.

For now, I’ll continue assuming the original.

✔ Answer (if correct): $ \boxed{24x^2 + 74x - 9} $

---

8. $ \frac{1}{2}(3(2x - 3)) + 3(4x + 2) - 6x $

Step-by-step:
- First simplify inside:
$$
3(2x - 3) = 6x - 9
$$
- Then:
$$
\frac{1}{2}(6x - 9) = 3x - \frac{9}{2}
$$
- Now add $ 3(4x + 2) = 12x + 6 $
- And subtract $ 6x $

Now combine all:
$$
3x - \frac{9}{2} + 12x + 6 - 6x
$$
Combine $ x $-terms:
$$
(3x + 12x - 6x) = 9x
$$
Constants:
$$
-\frac{9}{2} + 6 = -\frac{9}{2} + \frac{12}{2} = \frac{3}{2}
$$

✔ Answer: $ \boxed{9x + \frac{3}{2}} $

---

9. $ \frac{3}{4}(8x + 16) + 4(2x - 1) + 2(1) $

Step-by-step:
- $ \frac{3}{4}(8x) = 6x $, $ \frac{3}{4}(16) = 12 $
- $ 4(2x) = 8x $, $ 4(-1) = -4 $
- $ 2(1) = 2 $

Now sum:
$$
6x + 12 + 8x - 4 + 2 = (6x + 8x) + (12 - 4 + 2) = 14x + 10
$$

✔ Answer: $ \boxed{14x + 10} $

---

10. $ 6(2x + 3) - 3(2 - 4x) + 4(3x - 1) $

Step-by-step:
- Distribute:
$$
6(2x) + 6(3) -3(2) +3(4x) + 4(3x) - 4(1) = 12x + 18 - 6 + 12x + 12x - 4
$$
- Combine:
$$
(12x + 12x + 12x) + (18 - 6 - 4) = 36x + 8
$$

✔ Answer: $ \boxed{36x + 8} $

---

✔ Final Answers:

1. $ 11x - 2 $
2. $ 18x + 31 $
3. $ -26x + 63 $
4. $ -3x + 90 $
5. $ -22x + 8 $
6. $ \frac{46}{3}x - \frac{11}{6} $
7. $ 24x^2 + 74x - 9 $ (Quadratic — possibly typo)
8. $ 9x + \frac{3}{2} $
9. $ 14x + 10 $
10. $ 36x + 8 $

---

Let me know if you'd like me to double-check any specific problem or clarify any steps!