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The problem asks us to determine in which quadrant the solution of the given system of linear equations lies. The equations are:

1. \( x - 3y = 6 \)
2. \( x + y = 2 \)

### Step 1: Solving the System of Equations
We can solve the system using the substitution or elimination method. Let’s solve using substitution:

1. From the second equation:  
   \[
   x + y = 2 \implies y = 2 - x
   \]

2. Substitute \( y = 2 - x \) into the first equation:  
   \[
   x - 3(2 - x) = 6
   \]

3. Simplify the equation:  
   \[
   x - 6 + 3x = 6 \implies 4x - 6 = 6
   \]

4. Solving for \( x \):  
   \[
   4x = 12 \implies x = 3
   \]

5. Substitute \( x = 3 \) back into \( y = 2 - x \):  
   \[
   y = 2 - 3 = -1
   \]

### Step 2: Determining the Quadrant
The solution is \( (3, -1) \). In the coordinate plane:

- \( x = 3 \) is positive.
- \( y = -1 \) is negative.

This corresponds to **Quadrant IV**, where \( x > 0 \) and \( y < 0 \).

### Final Answer
The solution to the system of equations lies in **Quadrant IV**.

Would you like a detailed explanation of any step or have any follow-up questions?

### Related Questions
1. What method can be used to solve a system of linear equations besides substitution?
2. How can you determine the quadrant of a point without plotting it?
3. What happens when a system of linear equations has no solution?
4. How do you graphically find the intersection of two linear equations?
5. What do parallel lines in a system of equations signify?

**Tip**: Always check your solution by substituting the values back into the original equations.

The solution to the given system of linear equations lies in which quadrant? Solve: x - 3y = 6 and x + y = 2.

Learn how to solve the system of linear equations x - 3y = 6 and x + y = 2 and determine the quadrant of the solution. This example covers key algebraic concepts like substitution and solving systems of equations. Suitable for students in Grades 8-10.

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