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

Key Concepts

Set Notation

Set notation is a mathematical convention used to succinctly express a solution set. For a unique solution, it might be written as { (x, y) } for the ordered pair that satisfies the system; for no solution or infinitely many solutions, the notation clearly indicates the nature of the solution set, such as an empty set or a set containing all numbers in the case of infinite solutions.

Parallel and Coincident Lines

When graphing a system of equations, if the lines are parallel, they never intersect, which means the system has no solution. If the lines are exactly on top of each other (coincident), every point on the line is a solution, and the system has infinitely many solutions.

Intersection Point

The intersection point of two lines represents the unique solution to a system of linear equations when the lines cross at exactly one point. This point contains the values of the variables that satisfy both equations simultaneously.

Graphing Linear Equations

Graphing linear equations involves plotting each equation on a coordinate plane. By drawing the lines for each equation, one can visually identify the point of intersection, which represents the solution to the system, or determine if the lines are parallel (indicating no solution) or coincident (indicating infinitely many solutions).

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. Solving such a system means finding the values for the variables that satisfy all equations simultaneously, which often represents the intersection point of their graphs.

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