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.
$$
\left\{\begin{array}{rr}
x-y-z= & 1 \\
-x+2 y-3 z= & -4 \\
3 x-2 y-7 z= & 0
\end{array}\right.
$$

Key Concepts

Consistency and Inconsistency

The concepts of consistency and inconsistency refer to whether a system of equations has a solution. A consistent system has at least one solution, which can be either unique or infinitely many. An inconsistent system, on the other hand, has no solution because the equations represent constraints that cannot be simultaneously satisfied, often indicated by parallel lines or planes in a geometric interpretation.

Graphing Linear Equations

Graphing linear equations provides a visual method for understanding the solution to a system. In two dimensions, each equation represents a line and their intersections indicate the solution(s) to the system. In higher dimensions, the equations represent planes or hyperplanes. Graphing helps to illustrate whether the system has a unique intersection point, infinitely many intersections, or no intersection at all, clarifying the nature of the system's solutions.

System of Linear Equations

A system of linear equations consists of multiple linear equations that involve the same set of unknowns. These systems represent geometric objects such as lines, planes, or hyperplanes, and solving the system means finding the common intersection among these objects. The solution can be a unique point, infinitely many points (when the lines or planes coincide), or no points at all (when the objects are parallel or do not intersect).

Elimination Method

The elimination method is a standard technique for solving systems of equations by combining them in such a way that one or more variables are eliminated. This is typically done by adding or subtracting equations after scaling them appropriately, allowing you to reduce the system step-by-step until you can solve for the remaining variables. This method is especially useful for systems with three or more variables.

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