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.
$$
\left\{\begin{aligned}
x-y-z &=1 \\
2 x+3 y+z &=2 \\
3 x+2 y &=0
\end{aligned}\right.
$$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of equations where each equation represents a straight line (or plane in higher dimensions) and the solution to the system is the set of variable values that satisfy all equations simultaneously. This concept underpins many problems in algebra and geometry, where finding the intersection points of these lines (or the common intersection of planes) is the goal.

Consistent and Inconsistent Systems

A system of equations is termed consistent if there exists at least one set of values for the variables that satisfies each equation; otherwise, it is inconsistent if no such solution exists. Recognizing the consistency of a system is crucial as it determines whether the system has a unique solution, infinitely many solutions, or no solution at all.

Elimination and Substitution Methods

Elimination and substitution are two commonly used methods for solving systems of linear equations. The elimination method involves adding or subtracting equations to cancel out one or more variables, while the substitution method involves solving one equation for a variable and substituting that expression into the other equations, thus reducing the system step by step.

Graphical Representation of Equations

Graphing the equations in a system provides a visual method for understanding how the lines intersect, which corresponds to the solution of the system. Even though systems can extend beyond two variables, graphing is particularly useful in two-dimensional cases to illustrate concepts such as intersection points, parallel lines indicating no solution, or coinciding lines representing infinitely many solutions.

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