91Ó°ÊÓ

Linear algebra is a branch of mathematics that focuses on solving equations involving linear relationships.
In the context of systems of linear equations, we're dealing with equations that can be written in the form of lines in multidimensional space. These can be represented in matrix form, making it easier to apply methods for finding solutions.
The given problem is a system of linear equations:
1) \(2x - 4y + 6z = 12\)
2) \(6x - 12y + 18z = 36\)
3) \(-x + 2y - 3z = -6\)
Notice that linear algebra helps us to understand and solve these equations systematically. Let's explore some basic concepts that are related to solving such systems:

• Linear Equations: An equation that makes a straight line when it is graphed. In our case, each equation represents a plane in three-dimensional space.
• System of Linear Equations: A set of linear equations with the same variables. Here, we have three equations involving the variables \(x, y,\) and \(z\).
• Matrix Representation: These equations can also be represented in matrix form, which often simplifies the process of solving them.

By using linear algebra, we simplify our work and make it more systematic through well-defined methods and tools.

A parametric solution expresses the variables of a system as functions of one or more parameters.
In our problem, we ended up with the following system after simplification:
1) \(2x - 4y + 6z = 12\)
2) \(-x + 2y - 3z = -6\)
We used these steps:

• Identify Dependent Variables: We consider some variables as dependent and express them in terms of others. For example, solve for \(x\) from one equation.
• Introduce Parameters: We introduce parameters, \(t\) and \(s\), to represent \(y\) and \(z\).
• Express in Terms of Parameters: Substituting the parameterized values back in, we get the general solution.

In our example, let \(y = t\) and \(z = s\). Then, using \(x = 2y - 3z + 6\), we can write:

• \(x = 2t - 3s + 6\)
• \(y = t\)
• \(z = s\)

Therefore, the parametric form of the solution is:
\(x = 2t - 3s + 6\), \(y = t\), \(z = s\)
This gives us a clear, flexible way to represent all possible solutions of the system.

A system of linear equations can have infinite solutions when the equations are dependent.
That means they represent the same plane or line in space. In our case, we saw that:
1) \(2x - 4y + 6z = 12\)
2) \(6x - 12y + 18z = 36\)
Here, equation 2 is simply a multiple of equation 1. Hence, these two equations describe the same plane and do not add new information. This dependency suggests that the solutions are not unique; rather, there are infinitely many solutions.
To identify the infinite set of solutions, we use parameters (as shown earlier), to express each variable.
To summarize:

• Dependent Equations: Two or more equations that do not provide new information about the system.
• Consistent System: A system where all equations represent the same geometric entity.
• Infinite Solutions: There are countless solutions to the system, expressed in a parametric form.

This infinite nature is significant because it shows the flexibility in satisfying all equations in the system simultaneously. Through this method, students can get a deeper understanding of the relationships between variables in dependent systems.