91Ó°ÊÓ

When dealing with problems involving linear equations, we often encounter a system of equations that need to be analyzed and solved. In the context of intersecting planes, a system of linear equations can represent each plane by an individual equation. For instance, in 3-dimensional space, three linear equations will each act as a plane. The aim is to determine how these planes intersect—whether it is a point, a line, or not at all.

Consider the following system of equations, each representing a plane:

• First Plane: \(2x - y + z = 1\)
• Second Plane: \(x + y - z = -1\)
• Third Plane: \(-3x - 3y + 3z = 3\)

Initially, we can investigate dependency between these equations. A situation where one equation is a linear combination of others points to a dependent system. This often implies the planes intersect along a line in space rather than just at a single point. By examining their coefficients and relationships, we gain insights into the nature of their intersection.

Matrix row operations are practical tools to simplify and solve systems of linear equations. They help us express the system in a reduced form, making it easier to interpret relationships between variables. The goal is to transform the equations into a row-echelon form, where the system can be solved stepwise.

For the given system of equations, we represent it in an augmented matrix:\[\begin{bmatrix}2 & -1 & 1 & | & 1 \1 & 1 & -1 & | & -1 \-3 & -3 & 3 & | & 3\end{bmatrix}.\]
By using elementary row operations such as row addition, subtraction, and interchange, we aim to simplify the matrix. For example:

• Add 3 times the second row to the third row to eliminate redundant equations.
• Interchange the first and second rows to organize the matrix structure.

Eventually, achieving a matrix like:\[\begin{bmatrix}1 & 1 & -1 & | & -1 \2 & -1 & 1 & | & 1 \0 & 0 & 0 & | & 0\end{bmatrix}\]
indicates redundancy in the original equations and helps confirm that the intersecting planes indeed form a line.

A parametric solution is a way to express the infinite solutions of a system using one or more parameters. When planes intersect in a line, as in this scenario, it implies that the solution is not a single point but rather a continuous set of points along the line.

From the simplified matrix form, we derive two key equations:

• From \(x + y - z = -1\).
• From \(2x - y + z = 1\).

To find a parametric representation, assign a parameter, say \(z = t\). Substitute in the other equations to express \(x\) and \(y\) in terms of \(t\):

• \(y = -1 - x + t\).
• Substituting, solve for \(x\) and express \(y\) accordant to \(t\).

This leads to a parametric description of the line:
\[\begin{align*} x &= t, \ y &= -1 - 2t, \ z &= t,\end{align*}\]
where \(t\) represents any real number, depicting all points of intersection across the line formed by these planes.