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Let's analyze the system of linear equations in the table you provided. We will solve each system of equations, determine the type of solution, and classify the type of system (consistent, inconsistent, or dependent). Additionally, I'll describe how the graph of each system would appear.

### System 1:
\[
x - 3y = 2 \\
4x - 12y = 2
\]

1. **Type of System**:
   - The second equation is a multiple of the first equation (multiply the first equation by 4).
   - This means the two equations represent the same line, so the system is **dependent**.
   
2. **Graph**: 
   - The two lines are coincident, meaning they lie on top of each other.

3. **Solution**: 
   - Infinitely many solutions, as every point on the line is a solution.

---

### System 2:
\[
5x + 10y = 15 \\
10x + 5y = 5
\]

1. **Type of System**:
   - This system is neither proportional nor parallel at first glance. Let's solve it to find out.
   
   Simplify the first equation by dividing by 5:
   \[
   x + 2y = 3
   \]
   
   Simplify the second equation by dividing by 5:
   \[
   2x + y = 1
   \]
   
   Now, solve the system:
   
   - Multiply the second equation by 2 to align the coefficients of \(x\):
   \[
   2x + 4y = 6 \quad \text{and} \quad 2x + y = 1
   \]
   
   Subtract the second from the first:
   \[
   (2x + 4y) - (2x + y) = 6 - 1 \\
   3y = 5 \quad \Rightarrow \quad y = \frac{5}{3}
   \]
   
   Substitute \(y = \frac{5}{3}\) into the simplified first equation:
   \[
   x + 2\left(\frac{5}{3}\right) = 3 \\
   x + \frac{10}{3} = 3 \\
   x = 3 - \frac{10}{3} = \frac{9}{3} - \frac{10}{3} = -\frac{1}{3}
   \]
   
   Therefore, the solution is:
   \[
   x = -\frac{1}{3}, \quad y = \frac{5}{3}
   \]

2. **Type of System**:
   - **Consistent and independent**: This system has a single unique solution.

3. **Graph**:
   - The lines intersect at one point, which is \((-1/3, 5/3)\).

4. **Solution**:
   - \(\left(-\frac{1}{3}, \frac{5}{3}\right)\)

---

### System 3:
\[
-6x + 9y = -15 \\
2x - 3y = 5
\]

1. **Type of System**:
   - Simplify both equations:
     \[
     -6x + 9y = -15 \quad \Rightarrow \quad 2x - 3y = 5
     \]
     Multiply the second equation by 3:
     \[
     6x - 9y = -15
     \]
     Now add the equations together:
     \[
     (2x - 3y) + (-2x + 3y) = 5 + (-5) \\
     0 = 0
     \]
   
   This is a contradiction, indicating that the lines are parallel and do not intersect.

2. **Type of System**:
   - **Inconsistent**: No solution, since the lines are parallel.

3. **Graph**:
   - The lines are parallel and never intersect.

4. **Solution**:
   - No solution.

---

Would you like further clarifications or more detailed steps for any of these? 

### Relative questions:
1. What does it mean for a system to be dependent in linear equations?
2. How can you determine if two lines are parallel by just looking at their equations?
3. Can inconsistent systems occur with more than two equations?
4. How would you graphically represent a system with infinite solutions?
5. Why do systems of linear equations sometimes have no solution?

### Tip:
To easily check if two lines are parallel or coincident, compare their slopes after converting the equations to slope-intercept form (i.e., \( y = mx + b \)).

Complete the table of values for systems of linear equations.

Learn how to solve three systems of linear equations, including dependent, consistent, and inconsistent cases. This problem involves graphing the systems, analyzing their types, and finding their solutions. Suitable for students in Grades 9-12.

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