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

Verified step by step guidance

1. Start with the given system of equations: $x=9-2y$ and $x+2y=13$.
2. Since the first equation already expresses $x$ in terms of $y$, substitute $x=\; 9\; -\; 2y$ into the second equation.
3. After substitution, the second equation becomes $(9\; -\; 2y)\; +\; 2y\; =\; 13$. Simplify this equation by combining like terms.
4. Analyze the simplified equation to determine if it is true for all values of $y$, no values, or specific values. This will help identify if the system has one solution, no solution, or infinitely many solutions.
5. Based on the result, express the solution set using set notation. If there is one solution, write it as an ordered pair $(x,\; y)$. If no solution exists, state the empty set $\backslash varnothing$. If infinitely many solutions exist, express the solution set parametrically.

Key Concepts

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and graphing, each useful depending on the system's form.

Types of Solutions for Systems

Systems of linear equations can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent). Identifying the type involves analyzing the equations for contradictions or equivalences, which affects how the solution set is expressed.

Set Notation for Solution Sets

Set notation is a concise way to represent all solutions of a system. For a unique solution, it lists the ordered pair; for infinitely many solutions, it describes the solution set using parameters; for no solution, the set is empty. This notation clearly communicates the solution structure.