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

Verified step by step guidance

1. Start by writing down the system of equations: $x+3y=2$ and $3x+9y=6$.
2. Observe that the second equation is a multiple of the first equation. Specifically, multiply the first equation by 3: $3(x\; +\; 3y)\; =\; 3(2)$ which simplifies to $3x\; +\; 9y\; =\; 6$.
3. Since the second equation is exactly the same as the first equation multiplied by 3, the two equations represent the same line, meaning there are infinitely many solutions.
4. Express the solution set by solving the first equation for $x$ in terms of $y$: $x\; =\; 2\; -\; 3y$.
5. Write the solution set in set notation as $\{(x,\; y)\; |\; x\; =\; 2\; -\; 3y,\; y\; \backslash ,\; \backslash in\; \backslash ,\; \backslash mathbb\{R\}\}$, indicating all points on the line satisfy the system.

Key Concepts

Systems of Linear Equations

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

Methods for Solving Systems

Common methods to solve systems include substitution, elimination, and graphing. These techniques help find the point(s) where the equations intersect, representing the solution set. Choosing an appropriate method depends on the system's complexity and structure.

Types of Solutions and Set Notation

Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). Expressing solutions in set notation clearly defines the solution set, such as {(x, y) | x = 1, y = 0} for a unique solution or describing parameters for infinite solutions.