Let's solve the given system of equations step by step.

1. First equation: $4x - 16 = 32$ Solve for $x$: $4x = 32 + 16$ $4x = 48$ $x = \frac{48}{4} = 12$

2. Second equation: $3x + 9y = 27$ Substitute $x = 12$ from the first equation into this: $3(12) + 9y = 27$ $36 + 9y = 27$ $9y = 27 - 36$ $9y = -9$ $y = \frac{-9}{9} = -1$

3. Third equation: $2y + 6x = 18$ Substitute $x = 12$ and $y = -1$ into this equation: $2(-1) + 6(12) = 18$ $-2 + 72 = 18$ $70 \neq 18$ There is no solution for this system of equations because the third equation leads to a contradiction.

This means the system is inconsistent, and no common solution exists for all three equations.

Would you like more details, or do you have any questions?

Here are five related questions to expand on this:

1. How can we check if a system of equations is consistent before solving?
2. What does it mean for a system of equations to be inconsistent?
3. How can graphing the equations help visualize inconsistencies?
4. What methods exist for solving systems of equations (e.g., substitution, elimination)?
5. How would you approach solving a system with more than three equations?

Tip: When working with multiple equations, try checking for consistency early on by substituting known values into the remaining equations.