https://file.aiquickdraw.com/mathbot/file/1727754862302uw85cz05.jpg

The system of linear equations presented in the image is:

1. \( 2x + y - 3 = 0 \)
2. \( 4x + 2y = 12 \)

### Step-by-step solution:

#### 1. Rewrite the equations:
- From the first equation:
   \[
   2x + y = 3 \quad \text{(move 3 to the other side)}
   \]

   So the system becomes:
   \[
   2x + y = 3 \quad \text{(Equation 1)}
   \]
   \[
   4x + 2y = 12 \quad \text{(Equation 2)}
   \]

#### 2. Solve by substitution or elimination:
Let's use the **substitution method** by solving for \( y \) in Equation 1:
   \[
   y = 3 - 2x
   \]

#### 3. Substitute this into Equation 2:
   \[
   4x + 2(3 - 2x) = 12
   \]
   Simplifying:
   \[
   4x + 6 - 4x = 12
   \]
   \[
   6 = 12
   \]
This leads to a **contradiction**, which means there is **no solution** to this system. The lines are **parallel** and never intersect.

Thus, the system of equations is **inconsistent**, and there is **no solution**.

---

Would you like any further details or clarification on this?

Here are five related questions to consider:
1. What does it mean when two lines are parallel in terms of their slopes?
2. How would the solution change if the system had been consistent?
3. Can graphing these equations visually demonstrate that they are parallel?
4. What is the geometrical interpretation of an inconsistent system?
5. How can you solve other systems of equations using the elimination method?

**Tip:** If you reach a contradiction like \( 6 = 12 \), it typically indicates that the system of equations is inconsistent, meaning no solution exists.

Solve the system of equations: 2x + y - 3 = 0 and 4x + 2y = 12

Learn how to solve the system of linear equations 2x + y - 3 = 0 and 4x + 2y = 12 using substitution. This problem demonstrates the concept of inconsistent systems and parallel lines. Suitable for students in Grades 8-10.

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