Math Problem Statement

Give a geometric description of the following system of equations.

Solution

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

1st System:

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. --- 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!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Geometric Interpretation of Planes

Formulas

Plane equation in 3D: Ax + By + Cz = D

Theorems

Conditions for parallel planes
Conditions for intersecting planes
Linear dependence of equations

Suitable Grade Level

Undergraduate

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