Let's analyze the system of equations you provided:

1. \( y = x + 0(2) \)
   Simplifying \( 0(2) \), we get:
   \[
   y = x + 0 \implies y = x
   \]

2. \( y = x + 4(3) \)
   Simplifying \( 4(3) \), we get:
   \[
   y = x + 12
   \]

3. \( y = x - 4 \)

Now, we have the simplified system of equations:

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

To find the solution, let's compare them:

### 1. \( y = x \) and \( y = x + 12 \)
   Setting these equal to each other:
   \[
   x = x + 12
   \]
   This leads to a contradiction (no solution), meaning these two lines are parallel and never intersect.

### 2. \( y = x \) and \( y = x - 4 \)
   Setting these equal to each other:
   \[
   x = x - 4
   \]
   This leads to another contradiction, implying these lines are also parallel and never intersect.

### 3. \( y = x + 12 \) and \( y = x - 4 \)
   Setting these equal to each other:
   \[
   x + 12 = x - 4
   \]
   Solving for \( x \):
   \[
   12 = -4 \quad \text{(Contradiction again)}
   \]
   These lines are also parallel and do not intersect.

### Conclusion:
All three lines are parallel and there is no solution where all three equations meet.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the intersection of two linear equations?
2. What is the significance of parallel lines in a system of equations?
3. How would the system change if the slopes were different?
4. What does it mean when a system of equations has no solution?
5. Can parallel lines ever have a solution in a system of equations?

**Tip:** Parallel lines in a system of equations imply no solution because they never meet, having the same slope but different intercepts.

Explore how to solve a system of linear equations: y = x, y = x + 12, and y = x - 4. Understand why these equations represent parallel lines and why there is no solution. Learn about key algebraic concepts such as linear equations and systems of equations. Suitable for students in Grades 8-10.

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