Worksheet for simplifying linear expressions with algebraic equations and solutions.

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:

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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} $

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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} $

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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} $

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

Step-by-step:
- Distribute:
$$
4(5x) + 4(3) - 7(x) - 7(5) + 3(2x) - 3(3) = 20x + 12 - 7x - 35 + 6x - 9
$$
- Combine like terms:
$$
(20x - 7x + 6x) + (12 - 35 - 9) = 19x - 32
$$

✔ Answer: $ \boxed{19x - 32} $

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

Step-by-step:
- Distribute:
$$
\frac{2}{3}(6x) + \frac{2}{3}(2) - 3(x) - 3(2) + 2(3x) + 2(1)
= 4x + \frac{4}{3} - 3x - 6 + 6x + 2
$$
- Combine like terms:
$$
(4x - 3x + 6x) + \left(\frac{4}{3} - 6 + 2\right) = 7x + \left(\frac{4}{3} - 4\right)
= 7x + \left(\frac{4}{3} - \frac{12}{3}\right) = 7x - \frac{8}{3}
$$

✔ Answer: $ \boxed{7x - \frac{8}{3}} $

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

Step-by-step:
- Distribute:
$$
\frac{1}{2}x - 1 + \frac{2}{3}x - x
$$
- Combine like terms:
$$
\left(\frac{1}{2}x + \frac{2}{3}x - x\right) - 1
$$
- Find common denominator for coefficients:
$$
\frac{3}{6}x + \frac{4}{6}x - \frac{6}{6}x = \frac{1}{6}x
$$
- So:
$$
\frac{1}{6}x - 1
$$

✔ Answer: $ \boxed{\frac{1}{6}x - 1} $

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7. $ 4(3x + 4)(2y - 1) + 2(4xy + 2) $

Wait — this expression has two variables, $x$ and $y$, and includes a product of binomials.

But let's check: Is it $4(3x + 4)(2y - 1)$ or is there a typo?

Assuming it's meant to be simplified as written:

Step-by-step:

First, expand $ (3x + 4)(2y - 1) $:
$$
= 3x(2y) + 3x(-1) + 4(2y) + 4(-1) = 6xy - 3x + 8y - 4
$$

Now multiply by 4:
$$
4(6xy - 3x + 8y - 4) = 24xy - 12x + 32y - 16
$$

Now add $ 2(4xy + 2) = 8xy + 4 $

Add both parts:
$$
(24xy + 8xy) + (-12x) + 32y + (-16 + 4) = 32xy - 12x + 32y - 12
$$

✔ Answer: $ \boxed{32xy - 12x + 32y - 12} $

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

Again, two variables? But only $x$. Let's assume it's just $x$.

First, expand $ (x + 2)(2x - 1) $:
$$
= x(2x) + x(-1) + 2(2x) + 2(-1) = 2x^2 - x + 4x - 2 = 2x^2 + 3x - 2
$$

Multiply by $ \frac{1}{2} $:
$$
\frac{1}{2}(2x^2 + 3x - 2) = x^2 + \frac{3}{2}x - 1
$$

Now $ 3(6x + 3) = 18x + 9 $

Add:
$$
x^2 + \frac{3}{2}x - 1 + 18x + 9 = x^2 + \left(\frac{3}{2}x + 18x\right) + 8
= x^2 + \frac{39}{2}x + 8
$$

✔ Answer: $ \boxed{x^2 + \frac{39}{2}x + 8} $

Note: This is quadratic, not linear. If the original problem was supposed to be linear, double-check if there’s a typo.

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9. $ \frac{3}{4}(6x - 18) + 4(2x - 3) + 2x $

Step-by-step:
- Distribute:
$$
\frac{3}{4}(6x) - \frac{3}{4}(18) + 8x - 12 + 2x
= \frac{18}{4}x - \frac{54}{4} + 8x - 12 + 2x
= \frac{9}{2}x - 13.5 + 8x - 12 + 2x
$$
- Convert to fractions:
$$
\frac{9}{2}x + 8x + 2x = \frac{9}{2}x + \frac{16}{2}x + \frac{4}{2}x = \frac{29}{2}x
$$
- Constants: $-13.5 - 12 = -25.5 = -\frac{51}{2}$

✔ Answer: $ \boxed{\frac{29}{2}x - \frac{51}{2}} $

Or write as: $ \boxed{\frac{29x - 51}{2}} $

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10. $ 4(2x + 3) - 3(2x - 4) + 3(2x - 5) $

Step-by-step:
- Distribute:
$$
8x + 12 - 6x + 12 + 6x - 15
$$
- Combine like terms:
$$
(8x - 6x + 6x) + (12 + 12 - 15) = 8x + 9
$$

✔ Answer: $ \boxed{8x + 9} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ 11x - 2 $ |
| 2 | $ 18x + 31 $ |
| 3 | $ -26x + 63 $ |
| 4 | $ 19x - 32 $ |
| 5 | $ 7x - \frac{8}{3} $ |
| 6 | $ \frac{1}{6}x - 1 $ |
| 7 | $ 32xy - 12x + 32y - 12 $ |
| 8 | $ x^2 + \frac{39}{2}x + 8 $ |
| 9 | $ \frac{29}{2}x - \frac{51}{2} $ |
| 10 | $ 8x + 9 $ |

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Let me know if you'd like these rewritten in decimal form or if you want help checking any specific one!