Solve each system by graphing. Identify systems with no...

Solve each system by graphing. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
$$
\left\{\begin{array}{l}{3 x-y=4} \\ {6 x-2 y=4}\end{array}\right.
$$

Key Concepts

Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal in solving these systems is to find the common solution(s) that satisfy every equation simultaneously, which geographically corresponds to the intersection point(s) of the lines represented by the equations.

Graphical Methods

Graphical methods involve plotting each linear equation on a coordinate plane and observing the intersection point(s) of the lines. This visual approach helps in identifying whether the system has a unique solution (lines intersect at one point), no solution (lines are parallel and never intersect), or infinitely many solutions (lines are coincident, meaning they lie on top of one another).

Consistent, Inconsistent, and Dependent Systems

A system of linear equations can be classified as consistent if there is at least one solution, or inconsistent if there are no solutions. Within the consistent category, systems can be independent (with exactly one unique solution) or dependent (with infinitely many solutions, as the equations represent the same line).

Set Notation for Representing Solutions

Set notation provides a formal way to describe the solution set of a system of equations. For a unique solution, the notation typically denotes a specific ordered pair, while for infinitely many solutions, the notation often involves a parameter that represents the entire set of solutions. In cases of no solution, the notation indicates an empty set.

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