https://file.aiquickdraw.com/mathbot/file/1736898652194ipyh7pxk.jpg

Let's work through the problem step by step based on the uploaded image.

### Step-by-Step Explanation:

**The system of equations:**
\[
x + 2y = 6
\]
\[
2x + 4y = 12
\]

#### A. Graph the system:
When graphing, you'll notice that these equations represent the same line. This means the two equations are dependent and overlap entirely, resulting in infinitely many solutions. 

#### B. Notice about the pair of equations:
The second equation, \(2x + 4y = 12\), is just a multiple of the first equation, \(x + 2y = 6\). Specifically, the second equation is obtained by multiplying the first equation by 2. This means the two equations are essentially the same.

#### C. Fill in the blanks for elimination:
To use elimination:
1. Write the two equations:
   \[
   x + 2y = 6
   \]
   \[
   2x + 4y = 12
   \]

2. Multiply the first equation by -2 to align terms for elimination:
   \[
   -2(x + 2y) = -2(6)
   \]
   This simplifies to:
   \[
   -2x - 4y = -12
   \]

3. Add the two equations:
   \[
   (-2x - 4y) + (2x + 4y) = -12 + 12
   \]
   \[
   0 = 0
   \]
   The result \(0 = 0\) confirms the system has infinitely many solutions because the two equations are dependent.

#### D. Relation of the final equation to infinite solutions:
The final equation \(0 = 0\) signifies that the two equations are identical, and hence the system does not have a unique solution. Instead, every point on the line \(x + 2y = 6\) (or equivalently \(2x + 4y = 12\)) is a solution.

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### Five Related Questions:
1. How do you verify if two linear equations are equivalent?
2. What are the differences between systems with one solution, no solutions, and infinitely many solutions?
3. Can you solve this system by substitution instead of elimination?
4. How can you represent this system in matrix form?
5. How does graphing help confirm the nature of the solutions?

### Tip:
When checking for infinitely many solutions, always verify if one equation is a scalar multiple of the other.

The system of equations x + 2y = 6 and 2x + 4y = 12 has infinitely many solutions. Analyze and solve the system using elimination, and explain the relationship between the equations and their solutions.

Explore how to solve the system of equations x + 2y = 6 and 2x + 4y = 12 using the elimination method. Understand the concept of dependent systems and the conditions for infinitely many solutions. This explanation is ideal for Grades 8-10 students.

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