Solve each system using matrices. If there is no solution or...

Solve each system using matrices. If there is no solution or if there are infinitely many solutions and a system’s equations are dependent, so state.
$\left\{\begin{aligned} x+y &=1 \\ y+2 z &=-2 \\ 2 x-z &=0 \end{aligned}\right.$

Key Concepts

Consistency and Dependency

In the context of systems of linear equations, consistency refers to whether there exists at least one solution. An inconsistent system has no solution, typically due to contradictory equations. Conversely, if the equations are dependent (i.e., one equation can be derived from another), the system may have infinitely many solutions. Identifying dependency is crucial in determining the nature of the system's solutions.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by converting the system’s augmented matrix into row echelon form using elementary row operations. This process helps to determine the number of solutions by clarifying whether the system is consistent or inconsistent and whether it has a unique solution or infinitely many solutions.

Matrix Representation

Representing a system of linear equations using matrices involves organizing the coefficients of the variables into a coefficient matrix, the variables into a vector, and the constant terms into another vector. This representation simplifies the process of applying systematic solution methods such as Gaussian elimination or finding the inverse of the matrix, when applicable.

System of Linear Equations

A system of linear equations consists of several linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. Understanding how these systems behave (unique solution, no solution, or infinitely many solutions) is fundamental in linear algebra.

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