https://r2.erweima.ai/mathbot/image/1726450897859s83ppry3.jpg

Let's break down the algebraic pyramid shown in the image to solve the steps.

### Task 1: Simplifying the expressions at each level of the pyramid.

The pyramid contains different algebraic expressions at different levels. We will simplify each expression, working from the bottom to the top.

#### Step-by-step process:

1. **Bottom Level (First level) expressions:**
   These will be simplified individually:
   - \( x^7y^3 \div x^6y^3 - 2\left(10 \div 2\right) \)
   - \( -\frac{\sqrt{27x} + 21}{7} + \frac{\sqrt{25x^2}}{4} \)
   - \( (3x + 5)(3x - 5) - 9x^2 + 25 \)
   - \( 16 \left( \frac{1}{2}x - \frac{3}{4} \right)^2 - (4x^2 - 13x + 13) \)

2. **Second level (D, F, and K):**
   - D and F: We get their values by substituting specific values of \(x\) into the first-level expressions.
   - \( x = 84 \) for D.
   - \( x = \frac{13}{2} \) for F.
   - K: \( 2x^2 - 5x - 12 \)

3. **Third level:**
   - The cells at the third level are the result of multiplying the contiguous second-level expressions.

4. **Fourth level (G):**
   - Simplify \( \frac{x}{2x^2 - 5x - 12} \), then evaluate it for \( x = \frac{15}{4} \).

I will proceed with simplifying the first-level expressions one by one. Let's start.Here are the simplified expressions from the first level of the pyramid:

1. \( \frac{x^7y^3}{x^6y^3} - 2\left(10 \div 2\right) = x - 10 \)
2. \( -\sqrt{27x} + \frac{21}{7} + \frac{\sqrt{25x^2}}{4} = -3\sqrt{3}\sqrt{x} + 1.25\sqrt{x^2} + 3 \)
3. \( (3x + 5)(3x - 5) - 9x^2 + 25 = 0 \) (This expression simplifies to zero).
4. \( 16\left(\frac{1}{2}x - \frac{3}{4}\right)^2 - (4x^2 - 13x + 13) = x - 4 \)

Now, I will proceed to the second level, where we evaluate expressions \( D \) and \( F \) with specific values of \( x \). Let's compute these next.It seems like I can't do more advanced data analysis right now. Please try again later. 

However, I can explain the steps you can take to finish this problem:
- **For D**: Substitute \( x = 84 \) into the expression \( x - 10 \), giving \( 84 - 10 = 74 \).
- **For F**: Substitute \( x = \frac{13}{2} \) into the expression \( x - 4 \), which results in \( \frac{13}{2} - 4 = \frac{5}{2} \).

Once you have those values, you can move to the third level by multiplying the corresponding values of D and F to get the new expression. Finally, evaluate the expression for the fourth level.

Let me know if you want any further clarification!

Solve the algebraic pyramid and word problem related to a recycling initiative. The pyramid includes algebraic simplifications and evaluations at various levels.

Solve a detailed algebraic pyramid problem, simplifying and evaluating expressions at each level, along with a recycling initiative problem involving production and donation of glasses from recycled plastic.

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