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.
$$
\left\{\begin{array}{rr}
2 x+y & =-4 \\
-2 y+4 z & =0 \\
3 x & -2 z=-11
\end{array}\right.
$$

Key Concepts

Graphical Representation of Equations

Graphing is an important method for visualizing linear equations and systems of equations. Each equation can be represented as a line (or plane, in higher dimensions) and the solution to the system corresponds to the point or points where these graphs intersect. Graphical methods provide an intuitive understanding of the nature of the solutions and the relationships between the equations.

Substitution and Elimination Methods

Substitution and elimination are fundamental algebraic techniques for solving systems of equations. The substitution method involves solving one equation for one variable and then replacing that expression in the other equations. The elimination method requires combining equations in a way that cancels out one of the variables, making it easier to solve the remaining equation.

Consistency of Systems

The concept of consistency in a system of equations refers to whether the system has at least one solution. A consistent system has either a unique solution or infinitely many solutions, while an inconsistent system has no solution because the equations contradict each other.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy every equation simultaneously. Understanding systems is a key part of algebra and is used to model and solve real-life situations where several constraints are present.

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