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.
$$\left\{\begin{array}{l}
2 x+3 y=6 \\
4 x+6 y=12
\end{array}\right.$$

Key Concepts

Set Notation for Solution Sets

Set notation is a concise way to represent the set of solutions to a system of equations. When a system has infinitely many solutions, set notation allows you to express all possible solutions, typically in terms of a free variable. Using this notation clarifies the complete solution set in a clear and mathematically precise manner.

Dependent Systems

A dependent system occurs when the equations in the system are multiples of each other, resulting in the same line when graphed. This means every point on the line is a solution to the system, leading to an infinite number of solutions. Recognizing dependency is important to differentiate it from systems that have a unique solution or no solution.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solving these systems involves finding values for the variables that satisfy all of the equations simultaneously. This concept is fundamental in algebra and appears in various applications such as modeling real-world phenomena and optimization problems.

Graphical Solution of Equations

Graphing is a visual strategy used to solve systems of linear equations. By plotting each equation as a line on the coordinate plane, one can determine the solution by identifying the point or points where the lines intersect. This method not only provides the solution but also helps in understanding the relationship between the equations.

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