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.
$$\left\{\begin{array}{l}
y=2 x \\
y=-x+6
\end{array}\right.$$

Key Concepts

Intersection of Lines

The intersection point of the graphs of linear equations is crucial because it represents the set of variable values that satisfy all equations in the system. If the lines intersect at a single point, the system has a unique solution. If the lines are parallel (never intersect), then the system has no solution, and if they coincide (overlap completely), the system has infinitely many solutions.

System of Linear Equations

A system of linear equations involves two or more linear equations that are considered together because they share common variables. The goal is to find values for the variables that satisfy all the equations simultaneously. These systems can have a unique solution, no solution, or infinitely many solutions, depending on how the equations relate to each other.

Graphical Solution Method

The graphical solution method involves plotting each linear equation on the same coordinate plane. The point(s) where the graphs intersect represents the solution(s) to the system. This method provides a visual representation of the system and helps to intuitively understand the relationship between the equations.

Set Notation for Solutions

Set notation is a concise way to represent the solution set of a system. A unique solution is often represented as a set containing an ordered pair. For systems with no solution or infinitely many solutions, set notation is used to clearly express these conditions, indicating the absence of an intersection or the existence of multiple solutions.

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