https://file.aiquickdraw.com/mathbot/file/1728751721217zsdeh1bs.jpg

The problem asks for a geometric description of each system of equations. To understand this, let's analyze each system individually:

### 1st System:
\[
\begin{aligned}
2x + 4y - 6z &= -12 \\
-3x - 6y + 9z &= 18
\end{aligned}
\]
Both of these equations represent planes in 3D space. To find their geometric relation, let's simplify the second equation by dividing it by -3:

\[
x + 2y - 3z = -6
\]

Notice that this is exactly half of the first equation. So, these two equations represent the same plane. **Geometric description:** The system represents a single plane.

### 2nd System:
\[
\begin{aligned}
2x + 4y - 6z &= 12 \\
-x + 5y - 9z &= 1
\end{aligned}
\]
These two equations represent two different planes in 3D space. Since they are not multiples of each other, the planes intersect along a line. **Geometric description:** The system represents two planes that intersect in a line.

### 3rd System:
\[
\begin{aligned}
2x + 4y - 6z &= 12 \\
-3x - 6y + 9z &= 16
\end{aligned}
\]
Simplifying the second equation by dividing by -3:

\[
x + 2y - 3z = -\frac{16}{3}
\]

This is not a multiple of the first equation, and they do not have consistent solutions. These two planes are parallel but distinct. **Geometric description:** The system represents two parallel planes with no intersection.

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Would you like further details or clarification on any step?

Here are 5 questions you might want to explore next:
1. What is the geometric meaning of a system with no solution?
2. How can you visualize the intersection of two planes in 3D space?
3. Can a system of three planes have a single point of intersection?
4. What is the algebraic process for determining if two planes are parallel?
5. How do you find the intersection line between two planes?

**Tip:** To analyze systems geometrically, always look for proportionality between the equations to identify parallelism or coincident planes!

Give a geometric description of the following system of equations.

Explore the geometric interpretation of systems of equations in 3D space. This problem demonstrates how to identify parallel planes, planes that intersect along a line, and coincident planes. It is suitable for undergraduate students studying linear algebra.

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