In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x = 9-2y x + 2y = 13

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Solutions can be unique, non-existent, or infinite, depending on the relationships between the equations.

Types of Solutions

In a system of equations, there are three types of solutions: a unique solution (one point of intersection), no solution (parallel lines that never intersect), and infinitely many solutions (coincident lines that overlap completely). Understanding these types helps in analyzing the behavior of the equations involved.

Set Notation

Set notation is a mathematical way to describe a collection of objects, often used to express solution sets. For example, a unique solution can be represented as a single point (x, y), while infinitely many solutions can be expressed as a set of points that satisfy a particular equation, often using parameters to describe the relationship.