To solve the system of equations:

\[
\begin{aligned}
1.\ & x - y = 6 \\
2.\ & 2x + y = 3
\end{aligned}
\]

We can solve it using either substitution or elimination. I'll use the **substitution method**:

### Step 1: Solve the first equation for \(x\):
\[
x = y + 6
\]

### Step 2: Substitute \(x = y + 6\) into the second equation:
\[
2(y + 6) + y = 3
\]

### Step 3: Simplify and solve for \(y\):
\[
2y + 12 + y = 3 \\
3y + 12 = 3 \\
3y = 3 - 12 \\
3y = -9 \\
y = -3
\]

### Step 4: Substitute \(y = -3\) back into the expression for \(x\):
\[
x = -3 + 6 \\
x = 3
\]

### Final Answer:
The solution to the system is:
\[
x = 3, \quad y = -3
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you solve the system using the elimination method?
2. What happens if you graph both equations? Where do they intersect?
3. Can you express the solution as a vector?
4. What is the determinant of the coefficient matrix for this system?
5. How can you verify the solution?

**Tip:** When solving systems of equations, always check your solution by substituting the values back into the original equations.

Learn how to solve the system of equations x - y = 6 and 2x + y = 3 using the substitution method. This problem covers systems of linear equations and their solution methods. Suitable for high school students.

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