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. y = 3x - 521x - 35 = 7y

Verified step by step guidance

1. Step 1: Start by substituting the expression for y from the first equation into the second equation.
2. Step 2: Replace y in the second equation with (3x - 5) to get 21x - 35 = 7(3x - 5).
3. Step 3: Distribute the 7 on the right side of the equation: 21x - 35 = 21x - 35.
4. Step 4: Notice that both sides of the equation are identical, which means the equation is true for all values of x.
5. Step 5: Conclude that the system has infinitely many solutions, as both equations represent the same line.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. The solutions to these systems are the points where the equations intersect on a graph. Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). Understanding how to analyze these systems is crucial for determining their solution sets.

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 intersect at a single point. No solution arises when the lines are parallel and never meet, 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 represented as a single ordered pair (x, y), while no solution can be denoted by the empty set symbol ∅. Infinitely many solutions can be expressed using parameterization or interval notation, which helps in clearly communicating the nature of the solutions.