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.
$$\left\{\begin{array}{rr}
x+4 y-3 z= & -8 \\
3 x-y+3 z= & 12 \\
x+y+6 z= & 1
\end{array}\right.
$$

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations with the same set of variables. The goal is to find the variable values that satisfy all the equations simultaneously. These systems can be solved using various algebraic techniques, and understanding them is fundamental in linear algebra and its applications.

Methods of Solving Linear Systems

To solve a system of linear equations, techniques such as substitution and elimination are often used. Substitution involves isolating one variable in one equation and inserting the resulting expression into the other equations, while elimination involves combining equations to cancel out one or more variables. These methods help simplify complex systems into more manageable forms.

Consistency and Inconsistency

A key concept in solving systems of equations is determining whether the system is consistent or inconsistent. A consistent system has at least one solution, which means there is a set of variable values that satisfies all the equations. An inconsistent system, on the other hand, has no solution because the equations contradict one another.

Graphical Representation of Equations

Graphical representation involves plotting the equations on a coordinate system to visualize their relationships. For systems with two variables, the equations represent lines whose intersections indicate the solutions. Although graphing higher-dimensional systems (such as three-variable systems) is more complex, the concept helps in understanding the geometric interpretation of the solutions and the potential for consistency or inconsistency.

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