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 + 3y = 2 3x + 9y = 6

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), and infinitely many solutions (coincident lines). Identifying the type of solution is crucial for understanding the behavior of the system and can be determined through methods like substitution or elimination.

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 ordered pair (x, y), while infinitely many solutions can be expressed using parameters, such as { (x, y) | y = mx + b }, indicating a relationship between variables.