Solve each system of equations. If the system has no...

Solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems 19-30, graph the lines of the system.
$$
\left\{\begin{array}{r}
2 x-3 y-z=0 \\
-x+2 y+z=5 \\
3 x-4 y-z=1
\end{array}\right.
$$

Key Concepts

Consistency and Inconsistency

In the context of systems of equations, a system is said to be consistent if there is at least one set of variable values that satisfies all the equations; otherwise, the system is inconsistent. Determining consistency helps in understanding whether a unique solution, infinitely many solutions, or no solution exists, which is crucial in both theoretical studies and practical applications.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy every equation in the system simultaneously. Such systems occur in various fields and are fundamental to algebra and higher mathematics, serving as the basis for topics such as linear algebra and analytic geometry.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one or more variables. This method simplifies the system step-by-step, making it easier to solve for the remaining variables. It is widely used due to its systematic approach and applicability to systems of various sizes.

Graphical Interpretation of Solutions

Graphing the equations of a system provides a visual representation of the solution, where each equation is represented as a line (or plane in higher dimensions) and the intersection points of these graphs represent the solutions. This method aids in understanding the geometric relationships between the equations and can be particularly useful for visualizing scenarios where there is no intersection (inconsistency) or infinitely many intersections (dependent systems).

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