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.
$$\left\{\begin{array}{l}
x+y=5 \\
2 x+2 y=12
\end{array}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that share common variables. The goal is to identify values for the variables that satisfy all equations simultaneously. This concept is foundational in algebra and appears in various applications, from solving geometric problems to modeling real-world scenarios.

Graphical Solution Method

The graphical method for solving systems involves plotting each linear equation on a common coordinate plane. The point where the graphs intersect represents the solution to the system. This visual approach helps in understanding the relationship between equations and is particularly useful for identifying cases with a unique solution, no solution, or infinitely many solutions.

Inconsistent Systems

An inconsistent system is one in which the equations represent parallel lines that never intersect, indicating that there is no set of variable values that satisfies all equations simultaneously. Recognizing an inconsistent system is important because it reveals that the system has no solution, which must be clearly stated when solving the equations.

Set Notation for Solutions

Set notation is a concise way to express the solution of an equation or system. When a unique solution exists, it is written as a set containing the single solution pair. For an inconsistent system with no solution, the solution set is expressed as the empty set, and for systems with infinitely many solutions, the set includes all pairs that satisfy the given equations.

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