Trivial Solution Using Elimination by Addition

Solve the system of equations: $5x + 4y = 0$ and $2x - 2y = 0$, using elimination by addition to find the trivial solution.

When solving a system of linear equations, one of the powerful techniques we can utilize is elimination by addition. The goal of this method is to eliminate one of the variables by adding or subtracting the equations, thereby reducing the system to a simpler form, often a single equation in one variable. In this specific problem, the task is to find the trivial solution, which usually refers to the solution where all variables are zero. Such solutions are particularly relevant in the study of linear algebra, especially when considering homogeneous systems, where all constant terms are zero, leading to the origin as a solution point in linear space.

In exploring the elimination method, consider how the coefficients of the variables can be manipulated to facilitate the elimination of one variable. This involves multiplication of entire equations, if necessary, to create equal coefficients for one of the variables, so that they can be directly added or subtracted to cancel out, thus simplifying the task of solving for the other variable. Understanding this technique provides a solid foundation not only for solving equations by hand but also for comprehending the underlying structure of linear equations and their solutions geometrically, as lines or planes intersecting in space.

Additionally, connecting these concepts to vector spaces adds an enriched layer of understanding. Specifically, this problem can be related to the notion of linear independence where solutions to such problems indicate dependencies between equations. Homogeneous systems that have only the trivial solution highlight this independence, forming the basis of understanding more complex solutions that are explored further with notions like span, basis, and dimension in vector spaces.