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Let's solve the given system of linear equations using three methods: elimination, substitution, and Gaussian matrix operations with back substitution.

The system of equations given is:
\[ \begin{cases} 
3x + y = -12 \\
-2x + y = -2 
\end{cases} \]

### 1. Elimination Method

1. **Align the equations for elimination**: We can eliminate \( y \) by subtracting the second equation from the first:
   \[ (3x + y) - (-2x + y) = -12 - (-2) \]
   \[ 3x + 2x = -10 \]
   \[ 5x = -10 \]
   \[ x = -2 \]

2. **Substitute \( x \) back into one of the original equations**:
   \[ -2x + y = -2 \]
   \[ -2(-2) + y = -2 \]
   \[ 4 + y = -2 \]
   \[ y = -6 \]

### 2. Substitution Method

1. **Express \( y \) from the second equation**:
   \[ y = -2 + 2x \]

2. **Substitute into the first equation**:
   \[ 3x + (-2 + 2x) = -12 \]
   \[ 3x + 2x - 2 = -12 \]
   \[ 5x = -10 \]
   \[ x = -2 \]

3. **Find \( y \) using \( x = -2 \)**:
   \[ y = -2 + 2(-2) \]
   \[ y = -6 \]

### 3. Gaussian Matrix Operations

1. **Form the augmented matrix**:
   \[
   \begin{bmatrix}
   3 & 1 & |-12 \\
   -2 & 1 & |-2
   \end{bmatrix}
   \]

2. **Make the element below \( 3 \) zero**:
   \[ R_2 = R_2 + \frac{2}{3}R_1 \]
   \[
   \begin{bmatrix}
   3 & 1 & |-12 \\
   0 & \frac{5}{3} & |-\frac{26}{3}
   \end{bmatrix}
   \]

3. **Back substitution**:
   \[ \frac{5}{3}y = -\frac{26}{3} \]
   \[ y = -6 \]
   \[ 3x + 1(-6) = -12 \]
   \[ 3x = -6 \]
   \[ x = -2 \]

Hence, the solution for both \( x \) and \( y \) is \( x = -2 \), \( y = -6 \).

Would you like more details on any of these methods or have other questions?

**Further exploration questions:**
1. What happens if we change one of the equations slightly? How sensitive are the solutions?
2. Can we solve this system graphically? What would the graph look like?
3. What if the system had no solution or infinitely many solutions? How would the matrix method reveal that?
4. How would adding a third equation to the system affect the solution process?
5. What are some real-world applications of solving systems of linear equations?

**Tip:** When solving systems of equations, always check your solutions by substituting back into the original equations to ensure they satisfy all conditions.

Solve the following system of equations using ALL of the following methods:
 a) the elimination process
 b) the substitution process
 c) Gaussian matrix operations and back substitution

System of equations:
 3x + y = -12 
 -2x + y = -2

This page provides step-by-step solutions for solving a system of linear equations using the elimination method, substitution method, and Gaussian matrix operations with back substitution. The system is composed of two equations: 3x + y = -12 and -2x + y = -2. Suitable for college-level students.

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