If a system has an infinite number of solutions, use...

If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.
Solve using the elimination method.
$$
\begin{aligned}
&6 x-3 y=18\\
&6 x+3 y=-12
\end{aligned}
$$

Key Concepts

System of Linear Equations

This concept refers to a set of two or more equations that involve the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. Such systems can represent various geometric figures, like lines or curves, and their solutions correspond to the intersections of these figures.

Elimination Method

The elimination method is a technique used to solve systems of equations by combining the equations to cancel out one of the variables. This simplifies the system to a single equation with one variable, which can be solved easily. The value obtained is then substituted back into one of the original equations to find the other variable(s).

Types of Solutions in Linear Systems

Systems of linear equations can have a unique solution, no solution, or infinitely many solutions. A unique solution is found when the graphs of the equations intersect at one point, no solution arises when the graphs are parallel and never meet, and infinitely many solutions occur when the equations represent the same line.

Set-builder Notation

Set-builder notation provides a concise way to describe a set of elements that meet specific criteria. In the context of infinite solutions for a system of equations, it is used to specify all pairs (or tuples) of values that satisfy the given equation or condition, forming an entire solution set defined by a relationship between the variables.

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