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.
$$
\left\{\begin{array}{r}
x-y \quad=6 \\
2 x \quad-3 z=16 \\
2 y+z=4
\end{array}\right.
$$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. Systems can be solved using various methods, and their solutions represent the intersection of the graphs of the individual equations.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This reduces the number of variables in the other equations, simplifying the process of finding the solution to the system.

Elimination Method

The elimination method, also known as the addition method, involves adding or subtracting equations to eliminate one or more variables. By strategically combining equations, you can reduce the system step by step until you can solve for the remaining variables.

Consistency of Systems

A system of equations can be consistent or inconsistent. A consistent system has at least one solution, meaning that the graphs of the equations intersect at one or more points. An inconsistent system has no solution, which occurs when the equations represent parallel lines (or, in higher dimensions, non-intersecting planes) with no point in common.

Graphical Representation of Linear Equations

Graphing linear equations provides a visual interpretation of the solution to a system. Each equation is represented as a line (or plane, in higher dimensions), and the point or points where they intersect correspond to the solutions of the system. This method can help in understanding the relationships between the variables and in verifying the results obtained algebraically.

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