In Exercises 11-42, solve each system by graphing. If there...

In Exercises $11-42,$ solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}
x=2 \\
x=-1
\end{array}\right.$$

Key Concepts

Set Notation in Solution Sets

Set notation is a mathematical language used to express solution sets in a precise and compact form. For a system with a unique solution, the solution is typically written as a single ordered pair, while an empty solution set is represented by the empty set symbol (?) to indicate that no solution exists. Set notation helps formalize statements about the solutions of an equation or system of equations.

Inconsistent System

An inconsistent system occurs when the equations represent lines that do not meet; in other words, there is no point of intersection. Inconsistent systems have no solution, meaning that there is no set of variable values that can simultaneously satisfy all equations in the system.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations that share the same variables. The goal is to find the set of values that satisfy all equations in the system simultaneously. The solution may be a single point, an infinite set of points, or there can be no solution if the equations represent parallel lines that never intersect.

Graphical Solution Method

The graphical method involves plotting each equation on a coordinate plane and identifying the point or points where the graphs intersect. This visual approach helps illustrate the concept of intersection, which corresponds to the solution of the system, and can clearly show cases where there is no intersection (no solution) or infinitely many intersections (coincident lines).

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