The planes x+2 y+z=9 and 3 x+y-2 z=2 intersect in the line ℓwhose equation is (x, y, z)=(-1,5,0)

step 1 Okay, so in this problem, we're going to have to use conjectures to figure out the phone. So wha

The planes x+2 y+z=9 and 3 x+y-2 z=2 intersect in the line...

Key Concepts

Parametric Representation of Lines

A line in three-dimensional space can be described by a point through which it passes and a direction vector, with a scalar parameter accounting for the position along the line. This approach is useful because it provides a clear, algebraic way to represent all points on the line and is widely used for solving intersection problems and applying additional constraints.

Substitution Method

The substitution method is a strategy for solving systems of equations where one variable is expressed in terms of another and then substituted into the other equations. In problems involving additional constraints, such as setting a specific coordinate value, this method simplifies the process of identifying the corresponding parameter value that meets the condition.

Intersection of Planes

When two distinct non-parallel planes intersect in three-dimensional space, their intersection forms a line. Understanding this concept involves recognizing that the solution set of the two plane equations is a one-dimensional subset of ?³, usually expressed in parametric form. This idea is vital in linear algebra and geometry as it provides insights into the spatial relationships between different geometric figures.

Solving Systems of Linear Equations

Finding the intersection point or line of multiple geometric objects often requires solving a system of linear equations. In the context of intersecting planes, the simultaneous solution of the equations leads to either a unique point, a line of intersection, or indicates inconsistency. Mastering this concept is fundamental in linear algebra for resolving relationships among different variables.

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