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

Key Concepts

Graphical Interpretation of Linear Equations

Each linear equation can be represented graphically as a straight line in a two-dimensional plane, or as a plane in three dimensions. The graphical approach provides intuition, as the solution of a consistent system corresponds to the point or region where the graphs intersect. In problems that involve graphing, understanding the geometric representation can help visualize the relationships between the equations.

Consistency and Inconsistency

A system of equations is termed consistent if it has at least one solution, and inconsistent if it has no solutions. When solving a system, if the process of elimination or substitution leads to a contradiction, such as an equation that simplifies to a false statement, it indicates that the system is inconsistent.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find the values for the variables that satisfy all the equations simultaneously. This concept is foundational in algebra and appears in many scientific and engineering applications, where relationships between quantities are modeled with linear expressions.

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 step-by-step approach reduces the system to one with fewer variables, making it easier to solve for the remaining unknowns. It is particularly useful for systems with three or more equations.

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