Systems of Equations Flashcards

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Questions and Answers

What is a system of linear equations?

A single linear equation

An equation with one variable

A set of two or more linear equations containing two or more variables (correct)

Linear equations that have no solution

What is the solution of a system of linear equations?

An ordered pair that satisfies all the equations in the system.

What is the substitution method?

A method used to solve systems of equations by solving one equation for one variable and substituting into the other equation(s).

What is the elimination method?

A method used to solve systems of equations where one variable is eliminated by adding or subtracting the equations. Signup and view all the answers

What is a consistent system?

A system of equations or inequalities that has at least one solution. Signup and view all the answers

What is an inconsistent system?

A system of equations or inequalities that has no solution. Signup and view all the answers

What is an independent system?

A system of equations that has exactly one solution. Signup and view all the answers

What is a dependent system?

A system of equations that has infinitely many solutions. Signup and view all the answers

What is the solution of a linear inequality?

An ordered pair or ordered pairs that make the inequality true. Signup and view all the answers

What kind of system do parallel lines represent?

Inconsistent Signup and view all the answers

How many solutions do two lines with different slopes have?

1 Signup and view all the answers

What defines a consistent independent system?

Has exactly one solution Signup and view all the answers

Which method is NOT used for solving a system of equations?

Multiplication Signup and view all the answers

What are constraints in a system of inequalities?

Conditions given to variables, often expressed as inequalities. Signup and view all the answers

What is a feasible region?

The intersection of graphs in a system of constraints. Signup and view all the answers

What does linear programming involve?

Finding the maximum or minimum values of a function Signup and view all the answers

An unbounded feasible region is formed by a system of inequalities.

True Signup and view all the answers

What are the vertices of a feasible region?

Points located at the angles of a feasible region. Signup and view all the answers

Study Notes

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.

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Description

This quiz features a series of flashcards focused on systems of linear equations and inequalities. It includes definitions and examples of key concepts such as the solution of a system and the substitution method. Ideal for students preparing for tests in algebra.