We are given the following system of equations:

1. $4x + 3y + 2z = -6$
2. $x - 4y + 3z = -8$
3. $4x + y + 3z = -9$

We are asked to solve for $y$.

Step 1: Express $y$ from one of the equations

Take the third equation: $4x + y + 3z = -9$ Solve for $y$: $y = -9 - 4x - 3z$

Step 2: Substitute this value of $y$ into the other equations

Substitute $y = -9 - 4x - 3z$ into the first and second equations.

Substitution into the first equation:

$4x + 3(-9 - 4x - 3z) + 2z = -6$ Simplify: $4x - 27 - 12x - 9z + 2z = -6$ $-8x - 7z = 21$ This simplifies to: 8x + 7z = -21 \tag{4}

Substitution into the second equation:

$x - 4(-9 - 4x - 3z) + 3z = -8$ Simplify: $x + 36 + 16x + 12z + 3z = -8$ 17x + 15z = -44 \tag{5}

Step 3: Solve the system of two equations

Now we have two equations:

1. $8x + 7z = -21$
2. $17x + 15z = -44$

Solve this system for $x$ and $z$.

Multiply the first equation by 15 and the second equation by 7 to eliminate $z$:

$15(8x + 7z) = 15(-21) \implies 120x + 105z = -315$ $7(17x + 15z) = 7(-44) \implies 119x + 105z = -308$

Subtract the second equation from the first:

$(120x + 105z) - (119x + 105z) = -315 - (-308)$ $x = -7$

Step 4: Substitute $x = -7$ into one of the equations to find $z$

Substitute $x = -7$ into the equation $8x + 7z = -21$: $8(-7) + 7z = -21$ $-56 + 7z = -21$ $7z = 35$ $z = 5$

Step 5: Substitute $x = -7$ and $z = 5$ into the expression for $y$

Substitute $x = -7$ and $z = 5$ into $y = -9 - 4x - 3z$: $y = -9 - 4(-7) - 3(5)$ $y = -9 + 28 - 15$ $y = 4$

Thus, the value of $y$ is $\boxed{4}$.

Would you like more details on any step? Here are five related questions to explore:

1. How can you solve systems of linear equations using matrix methods?
2. What are the graphical interpretations of solving linear systems?
3. How does substitution differ from elimination when solving equations?
4. What is Cramer's rule for solving systems of linear equations?
5. How can you apply these methods to non-linear systems?

Tip: Always double-check your solutions by substituting back into the original equations to verify accuracy.