Systems of Equations Flashcards

Systems of Linear Equations

• Comprised of two or more linear equations with multiple variables, e.g., 2x + 3y = -1 and x - 3y = 4.
• Solutions include ordered pairs that satisfy all equations within the system, such as (1, -2) for x + y = -1 and -x + y = -3.

Methods for Solving Systems

• Substitution Method: Solve one equation for a variable, substitute into others, and solve sequentially (example results in (5, 10)).
• Elimination Method: Combine equations to eliminate a variable, then solve the remaining variables (example leads to solution (-3, 8)).

Types of Systems

• Consistent System: At least one solution exists, e.g., x + y = 6 and x - y = 4 yielding (5, 1).
• Inconsistent System: No solutions exist, indicated by parallel lines.
• Independent System: Exactly one solution is present, e.g., x + y = 7 and x - y = 1 with solution (4, 3).
• Dependent System: Infinitely many solutions, examples include x + y = 2 and 2x + 2y = 4.

Solutions of Inequalities

• Solutions represent ordered pairs making inequalities true, e.g., 3x + 2y ≥ 6.
• A set of linear inequalities forms a system, such as 2x + 3y > -1 and x - 3y ≤ 4.

Graphical Representation

• Coinciding Lines: Represent a consistent and dependent system with infinite solutions (same slope and intercept).
• Intersecting Lines: Indicate a consistent and independent system with one solution (different slopes).
• Parallel Lines: Show an inconsistent system with zero solutions (same slope, different intercepts).

Solutions and Characteristics

• Systems of equations can yield:
  • One solution for intersecting lines (distinct slopes).
  • No solutions for parallel lines (same slope, distinct y-intercepts).
  • Infinite solutions for coinciding lines (identical slopes and intercepts).

Situational Attributes of Systems

• Consistent systems are characterized by coinciding or intersecting lines.
• Inconsistent systems are defined by parallel lines.
• Graphs indicating all points of intersection denote consistent systems.

Solving Techniques

• Methods include graphing, substitution, and elimination to find solutions effectively.
• Constraints in a system are often represented as linear inequalities, defining the relationships between variables.

Feasible Regions in Linear Programming

• Feasible Region: Intersection of graphs defined by constraints.
• Bounded Region: A closed region created by inequalities.
• Unbounded Region: An open region with no limits defined by inequalities.

Relevant Terminology

• Vertices of a Feasible Region: Points at the corners or angles of the feasible region where constraints meet.
• Linear programming optimizes functions within defined constraints, aiming for maximum or minimum values.