How to Solve the System of Equations: x2y3z 0, 2x4y6z 0, 3x6y9z 0

These three equations, while seemingly different, all represent the same plane in three-dimensional space. Understanding this concept is crucial for grasping the solution space of such systems. Let's explore the detailed steps to solve this system and why there isn't a unique solution.

Understanding the Equations

Consider the given system of equations:

x2y3z 0

2x4y6z 0

3x6y9z 0

If we multiply the first equation by 2, we get the second equation. If we multiply the first equation by 3, we derive the third equation. Therefore, all these equations represent the same plane in three-dimensional space.

Visualization and Interpretation

The set of all points (x, y, z) that satisfy these equations forms a plane. Every point on this plane is a solution to the system. This plane can be described as:

({x, y, z} | x -2y - 3z)

We can also write this plane as:

({ -2y - 3z, y, z } | y and z are real numbers)

This expression denotes that the plane is defined by the parameters y and z, with x being a function of these two parameters.

Parameterization of the Solution Space

Let's parameterize the solution space using two parameters, t and s, where:

y t

z s

From the first equation, we can express x as:

x -2y - 3z -2t - 3s

Thus, the general solution can be written as:

({x, y, z} { -2t - 3s, t, s } | t, s in R)

This means that there are infinitely many solutions, depending on the values of t and s.

Conclusion

The solution space for the given system of equations is a plane in three-dimensional space. Understanding this plane is key to deciphering the system's behavior. Each point on this plane is a solution, and there is no unique point (x0, y0, z0) that is the only solution.