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.
$$
\left\{\begin{array}{l}
3 x-2 y+2 z=6 \\
7 x-3 y+2 z=-1 \\
2 x-3 y+4 z=0
\end{array}\right.
$$

Key Concepts

Elimination Method

The elimination method is a systematic approach to solving systems of linear equations by eliminating one variable at a time. This is achieved through adding or subtracting multiples of the equations from each other so that one variable cancels out, reducing the system to fewer equations. The process is repeated until the system is reduced to a single equation in one variable, which can then be solved using back substitution.

Graphical Representation of Linear Systems

Graphical representation involves plotting each equation on a coordinate system to visually identify the point(s) at which the graphs intersect. For two-variable systems, this typically results in lines on a plane, where their intersection point represents the solution. Although graphing becomes more complex for higher-dimensional systems, understanding the graphical interpretation is crucial for visualizing the concept of consistency and the nature of the solutions.

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 values for the variables that satisfy all equations simultaneously. These systems can be represented in various forms, such as algebraic equations, matrices, or graphs, and they form the basis for understanding more complex relationships in mathematics and applied sciences.

Consistency of a System

A system of equations is said to be consistent if there exists at least one solution that satisfies all the equations; if not, the system is inconsistent. Systems can have a unique solution (consistent and independent), infinitely many solutions (consistent and dependent), or no solution at all (inconsistent). Recognizing the consistency of a system is fundamental, as it determines whether a solvable intersection of the equations exists.

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