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.
$$\left\{\begin{array}{l}
x-2 y=4 \\
2 x-4 y=8
\end{array}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations involving the same set of variables. The goal is to find a set of values that simultaneously satisfies all the equations. Such systems can represent geometric objects, like lines in the plane, where the solution represents the point of intersection.

Graphical Method

The graphical method involves converting each linear equation into a visual representation on the coordinate plane, typically by rewriting each equation in slope-intercept form. By plotting the lines corresponding to each equation, you can observe the point of intersection, if any, which corresponds to the solution of the system.

Slope-Intercept Form

Rewriting a linear equation in slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept, allows for straightforward graphing and comparison of lines. This form makes it easy to see how lines relate to one another in terms of steepness and position, which is crucial for identifying intersections, parallelism, or overlapping lines.

Consistency and Dependence in Linear Systems

A linear system can be classified as consistent if it has at least one solution and inconsistent if it has none. Within consistent systems, if the equations represent the same line, the system is dependent and has infinitely many solutions. Conversely, if they intersect at a single point, the system is independent and has a unique solution.

Set Notation for Solution Sets

Set notation provides a precise way of expressing the answer to a system of equations. When a system has a unique solution, the solution can be written as a set containing the ordered pair of values. For systems with no solution or infinitely many solutions, set notation allows us to clearly indicate these outcomes, either by stating that no ordered pair exists or by describing the solution in terms of a parameter.

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