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To solve this problem, let's analyze the original system of equations given:

\[
5x + 4y = 35
\]
\[
4x + 2y = 16
\]

We need to identify an equivalent system of equations. To do this, we can manipulate the equations by applying operations like multiplication, subtraction, or adding terms, to see which option matches the transformed equations while maintaining the same solutions.

### Step 1: Simplify the Second Equation
Let's simplify the second equation, \( 4x + 2y = 16 \), by dividing all terms by 2:

\[
2x + y = 8
\]

Now we have an equivalent but simplified system:
1. \( 5x + 4y = 35 \)
2. \( 2x + y = 8 \)

### Step 2: Review the Answer Choices
Let's examine each answer choice to see which one matches or could be derived from the simplified system.

1. **Option 1:** 
   \[
   5x + 4y = 35
   \]
   \[
   -8x - 4y = -32
   \]

2. **Option 2:**
   \[
   5x + 4y = 35
   \]
   \[
   -8x - 4y = 16
   \]

3. **Option 3:**
   \[
   5x + 4y = 35
   \]
   \[
   4x - 4y = -32
   \]

4. **Option 4:**
   \[
   5x + 4y = 35
   \]
   \[
   -8x + 2y = -32
   \]

### Step 3: Identify the Correct Choice
We are looking for an option where one of the equations is equivalent to either the original or simplified forms of \(5x + 4y = 35\) and \(2x + y = 8\). 

Analyzing each:

- **Option 1**: By dividing the second equation \(-8x - 4y = -32\) by \(-4\), we get \(2x + y = 8\), which matches the simplified form of the original system.  
  This option matches the system of equations we derived.

Therefore, **Option 1** is correct.

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Would you like more details or have any questions?

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Here are some related questions for practice:

1. How can you check if two systems of equations are equivalent?
2. What happens to the solution of a system if you multiply one equation by a non-zero constant?
3. How can substitution help in solving a system of equations?
4. How does elimination differ from substitution in solving systems of equations?
5. Why is it important to simplify equations when solving a system?

**Tip:** When solving systems of equations, try simplifying or rewriting equations to see if they match any of the answer choices in a different form.

Which system of equations has the same solution as the system below?

5x + 4y = 35
4x + 2y = 16

Identify the equivalent system of equations for 5x + 4y = 35 and 4x + 2y = 16. This problem involves algebraic techniques for finding equivalent systems through equation manipulation. Ideal for students in Grades 8-10, this example covers key concepts in solving systems of equations.

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