If there is no solution or if there are infinitely many...

If there is no solution or if there are infinitely many solutions and a system's equations are dependent, so state.
$$\left\{\begin{array}{rr}x-y+3 z= & 8 \\3 x+y-2 z= & -2 \\2 x+4 y+z= & 0\end{array}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to determine the values of these variables that satisfy all the equations simultaneously, which can be interpreted geometrically as finding the intersection of hyperplanes in multidimensional space.

Consistency of a System

The concept of consistency in a system of linear equations refers to whether the system has at least one solution. A system is considered consistent if there exists one or more solutions, and inconsistent if no solution exists. Determining consistency is essential in understanding the nature of the solution set.

Dependent Equations

Dependent equations occur when one or more equations in the system can be expressed as a linear combination of the others. This redundancy means the equations do not provide unique information about the solution set, which can lead to scenarios where the system has infinitely many solutions.

Infinitely Many Solutions

When a system of linear equations is underdetermined—often due to dependency among the equations—it may have an infinite number of solutions. In this case, some variables are free parameters, and the general solution is expressed in terms of these parameters, representing a continuous family of solutions.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations by performing row operations to reduce the system to a simpler form, typically the row-echelon form. This method helps in identifying the consistency of the system, as well as detecting dependencies among the equations.

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