Sure, let's solve the given system of linear equations step-by-step. The system is:
[tex]\[
\begin{cases}
x + 4y = -2 \\
3x + 2y = -6
\end{cases}
\][/tex]
To solve this system by elimination or substitution, let's follow these steps:
### Step 1: Start with the two given equations.
[tex]\[
\begin{cases}
x + 4y = -2 \quad \text{(1)} \\
3x + 2y = -6 \quad \text{(2)}
\end{cases}
\][/tex]
### Step 2: Multiply equation (1) to align coefficients for elimination.
Let's multiply equation (1) by 3 so that the coefficients of [tex]\( x \)[/tex] in both equations will be aligned form which we can easily eliminate [tex]\( x \)[/tex]:
[tex]\[
3(x + 4y) = 3(-2)
\][/tex]
This simplifies to:
[tex]\[
3x + 12y = -6 \quad \text{(3)}
\][/tex]
### Step 3: Subtract equation (2) from this new equation (3).
Now we have:
[tex]\[
\begin{cases}
3x + 12y = -6 \quad \text{(3)} \\
3x + 2y = -6 \quad \text{(2)}
\end{cases}
\][/tex]
Subtract the second equation from the first one:
[tex]\[
(3x + 12y) - (3x + 2y) = -6 - (-6)
\][/tex]
This simplifies to:
[tex]\[
10y = 0
\][/tex]
### Step 4: Solve for [tex]\( y \)[/tex].
Since [tex]\( 10y = 0 \)[/tex]:
[tex]\[
y = 0
\][/tex]
### Step 5: Substitute [tex]\( y \)[/tex] back into one of the original equations to solve for [tex]\( x \)[/tex].
Let's substitute [tex]\( y = 0 \)[/tex] into equation (1):
[tex]\[
x + 4(0) = -2
\][/tex]
This simplifies to:
[tex]\[
x = -2
\][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[
\boxed{x = -2, y = 0}
\][/tex]
