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.
$$\left\{\begin{array}{l}
x+y=2 \\
x-y=4
\end{array}\right.$$

Key Concepts

Set Notation for Expressing Solutions

Set notation provides a precise way to express the solution set of a system of equations. It involves enclosing the solution within braces and using ordered pairs for unique solutions. If the system does not have a unique solution, set notation can also be used to indicate either no solution, often denoted by an empty set, or infinitely many solutions, typically described by a line or a parameterized form.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to the system is a set of values that satisfy all of the equations simultaneously. This concept is foundational in algebra and forms the basis for methods of finding intersections between different linear relationships.

Graphical Method for Solving Systems

The graphical method involves plotting each equation on a coordinate plane and identifying the point or points at which the graphs intersect. This method provides a visual representation of the solution set and helps in understanding the relationship between the equations. It is particularly useful for illustrating whether a system has a unique solution, infinitely many solutions, or no solution at all.

Intersection Point as the Unique Solution

When two lines intersect at a single point, this point represents the unique solution to the system of equations. The coordinates of the intersection point satisfy both equations simultaneously, indicating that they are the exact values of the variables that make both equations true.

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