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.
$$
\begin{array}{r}
{2 x+y=6} \\
{x-y=3}
\end{array}
$$

Key Concepts

Types of Solutions in Linear Systems

Linear systems can be classified based on their solution sets: a unique solution exists when the graphs of the equations intersect at a single point; infinitely many solutions occur when the equations are dependent, meaning they represent the same line; and no solution exists when the equations are inconsistent, typically representing parallel lines that never meet.

Systems of Linear Equations

A system of linear equations is a collection of equations involving the same set of variables where each equation is linear. The goal is to find values for the variables that satisfy all equations simultaneously. Depending on the relationships among the equations, the system can have a unique solution, infinitely many solutions, or no solution at all.

Elimination Method

The elimination method is a technique for solving systems of linear equations by adding or subtracting equations to eliminate one of the variables. This process simplifies the system, making it easier to solve for the remaining variables step by step.

Set-builder Notation

Set-builder notation is a method for describing a set by stating the properties that its elements must satisfy. In the context of linear systems with infinitely many solutions, it is used to express the solution set in a compact form by introducing a parameter that represents the free variable.

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