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.
$$\left\{\begin{array}{l}x+y+2 z=11 \\x+y+3 z=14 \\x+2 y-z=5\end{array}\right.$$

Key Concepts

Systems of Linear Equations

Systems of linear equations consist of multiple equations with several unknowns, and they are used to find common solutions that satisfy all equations simultaneously. These systems can be analyzed using algebraic methods, and the nature of the solutions – whether unique, none, or infinitely many – depends on the relationships between the equations.

Consistency and Inconsistency

A system is consistent when at least one set of values for the unknowns satisfies all the equations simultaneously. In contrast, an inconsistent system has no solution because the equations contradict one another. Determining the consistency of a system is a fundamental step in analyzing linear systems.

Dependent Equations

Dependent equations are those that can be derived from one another through linear combinations, meaning they do not provide unique information about the variables. When a system includes dependent equations, it may lead to an underdetermined system where the solutions are not unique, possibly resulting in infinitely many solutions.

Infinitely Many Solutions

When a system has infinitely many solutions, it indicates that there are free parameters in the solution set. This situation typically arises in consistent systems that are underdetermined due to dependency among equations, thereby allowing for a continuum of solutions rather than a unique solution.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by performing row operations to transform the augmented matrix into a row echelon form or reduced row echelon form. This method helps in identifying whether a system is consistent, whether the equations are independent or dependent, and in finding the unique solution or describing the solution set when infinitely many solutions exist.

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