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The given system of equations is:

\[
\begin{aligned}
6d + 4.5e &= 16.5 \\
5d + 0.5e &= 4
\end{aligned}
\]

We need to find which of the provided systems are equivalent to this system. Two systems are considered equivalent if one can be transformed into the other by operations like multiplying or dividing equations by constants or rearranging terms.

### Step 1: Solve the original system of equations.

We can solve this system using either substitution or elimination. Let's use the elimination method:

1. Multiply the second equation by 9 to eliminate the decimals:
   \[
   9(5d + 0.5e) = 9(4)
   \]
   This gives:
   \[
   45d + 4.5e = 36
   \]

2. Now subtract the first equation from the scaled second equation:
   \[
   (45d + 4.5e) - (6d + 4.5e) = 36 - 16.5
   \]
   Simplifying:
   \[
   39d = 19.5
   \]
   Solving for \(d\):
   \[
   d = \frac{19.5}{39} = 0.5
   \]

3. Substitute \(d = 0.5\) into the second original equation:
   \[
   5(0.5) + 0.5e = 4
   \]
   Simplifying:
   \[
   2.5 + 0.5e = 4
   \]
   \[
   0.5e = 1.5 \quad \Rightarrow \quad e = \frac{1.5}{0.5} = 3
   \]

Thus, the solution to the original system is \(d = 0.5\) and \(e = 3\).

### Step 2: Check equivalent systems.

- **First system:** 
  \[
  6d + 4.5e = 16.5, \quad 45d + 4.5e = 4
  \]
  This is **not equivalent**, since the second equation should match one of the original forms when scaled.

- **Second system:**
  \[
  30d + 22.5e = 82.5, \quad 5d + 0.5e = 4
  \]
  The first equation is obtained by multiplying the first original equation by 5, and the second equation is unchanged. This system is **equivalent**.

- **Third system:**
  \[
  30d + 22.5e = 82.5, \quad 30d + 3e = 24
  \]
  This system is **not equivalent** because the second equation is incorrect.

- **Fourth system:**
  \[
  6d + 4.5e = 16.5, \quad 6d + 0.6e = 4.8
  \]
  This system is **not equivalent** because the second equation is incorrect.

- **Fifth system:**
  \[
  12d + 9e = 33, \quad 10d + 0.5e = 8
  \]
  The first equation is obtained by multiplying the first original equation by 2, and the second equation is scaled. This system is **equivalent**.

- **Sixth system:**
  \[
  6d + 4.5e = 16.5, \quad 11d + 5e = 20.5
  \]
  This system is **not equivalent** because the second equation is incorrect.

### Conclusion:
The systems that are equivalent to the given system are:
- Second system: \(\{30d + 22.5e = 82.5, 5d + 0.5e = 4\}\)
- Fifth system: \(\{12d + 9e = 33, 10d + 0.5e = 8\}\)

Would you like more details on the steps, or do you have any questions?

---

Here are 5 related questions to expand your understanding:

1. How can we use the substitution method to solve this system instead?
2. What does it mean for two systems of equations to be equivalent in linear algebra?
3. Can you generalize how to scale equations without changing the solution?
4. How can matrix methods (like Gaussian elimination) solve such systems?
5. How would the process change if one equation were nonlinear?

**Tip:** Always check your final solution by plugging the values back into both equations to verify correctness.

Select all systems that are equivalent to this system:
6d + 4.5e = 16.5
5d + 0.5e = 4

Learn how to solve a system of linear equations and identify equivalent systems. The given system involves equations with two variables, and you will explore which transformations yield equivalent forms. Suitable for students in Grades 8-10.

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