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.
$$
\begin{aligned}
&12 x-6 y=-15\\
&-4 x+2 y=5
\end{aligned}
$$

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 aim is to find all values of the variables that satisfy every equation simultaneously, which can lead to a unique solution, infinitely many solutions, or no solution at all depending on the relationship between the equations.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by combining equations to eliminate one of the variables. This process simplifies the system, often reducing it to a single equation with one variable, making it easier to solve for the remaining unknowns.

Infinite Solution Sets

An infinite solution set occurs when the equations in a system are dependent, meaning one equation can be derived from the other(s). This dependency indicates that the system represents the same geometric object (like the same line in two dimensions), leading to infinitely many solutions that satisfy every equation in the system.

Set-Builder Notation

Set-builder notation is a concise way of describing a set of elements that satisfy a particular property or condition. In the context of systems of equations, it is often used to express the solution set in terms of one or more parameters when the system has infinitely many solutions.

No Solution (Inconsistency)

A system has no solution when the equations are inconsistent, meaning they represent parallel lines or conflicting relationships that never intersect. This inconsistency implies that there are no variable values that can simultaneously satisfy all equations in the system.

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