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/4 - y/4 = −1 x + 4y = -9

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The solution to the system is the set of values that satisfy all equations simultaneously. Systems can be classified as consistent (having at least one solution) or inconsistent (having no solutions). Understanding how to manipulate and analyze these equations is crucial for finding solutions.

Types of Solutions

In the context of systems of equations, solutions can be classified into three categories: a unique solution, no solution, and infinitely many solutions. A unique solution occurs when the lines represented by the equations intersect at a single point. No solution arises when the lines are parallel, while infinitely many solutions occur when the equations represent the same line, leading to an infinite number of intersection points.

Set Notation

Set notation is a mathematical way to describe a collection of objects, often used to express the solution sets of equations. For example, a unique solution can be expressed as a single ordered pair (x, y), while infinitely many solutions may be represented using parameterization or interval notation. Understanding how to use set notation is essential for clearly communicating the nature of the solutions found in a system of equations.