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A system of linear equations in three variables has 1 solution if it contains two parallel lines. True O False

A system of linear equations in three variables has 1 solution if it contains two parallel lines. True O False

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Understanding Systems of Linear Equations in Three Variables A system of linear equations in three variables can have various types of solutions depending on the nature of the equations and their geometrical significance. Let's delve into one such scenario through a question: ### Question A system of linear equations in three variables has 1 solution if it contains two parallel lines. - **True** - **False** ### Analysis To understand the solution to this question, let's break down the components: 1. **System of Linear Equations in Three Variables:** A general form of such a system can be represented as: - \( a_1x + b_1y + c_1z = d_1 \) - \( a_2x + b_2y + c_2z = d_2 \) - \( a_3x + b_3y + c_3z = d_3 \) 2. **Parallel Lines:** In a three-dimensional coordinate system, two lines are considered parallel if they never intersect and have the same direction vector (slope in 2D). ### Graphical Representation In the context of linear algebra and three-dimensional geometry: - Parallel lines imply that there is no intersection or point common between them. - For a system to have a unique solution, the equations (or geometrical representations such as planes or lines) must intersect at a single point. ### Conclusion Given the nature of parallel lines, they do not meet. Hence, a system of linear equations in three variables that includes two parallel lines cannot have a unique solution (1 solution) because the parallel lines inherently don't provide a single point of intersection. ### Answer - **False** ### Further Explanation In a system involving three dimensions (x, y, z), for a unique solution to exist, the three planes represented by the equations must intersect at exactly one point. Involving two parallel lines violates this condition, leading to either no solution or infinitely many solutions if the entire third plane aligns with one of the existing lines. Understanding these geometric relationships helps us interpret the behavior of equations and their solutions in higher dimensions, which is vital in linear algebra and its applications.
Transcribed Image Text:### Understanding Systems of Linear Equations in Three Variables A system of linear equations in three variables can have various types of solutions depending on the nature of the equations and their geometrical significance. Let's delve into one such scenario through a question: ### Question A system of linear equations in three variables has 1 solution if it contains two parallel lines. - **True** - **False** ### Analysis To understand the solution to this question, let's break down the components: 1. **System of Linear Equations in Three Variables:** A general form of such a system can be represented as: - \( a_1x + b_1y + c_1z = d_1 \) - \( a_2x + b_2y + c_2z = d_2 \) - \( a_3x + b_3y + c_3z = d_3 \) 2. **Parallel Lines:** In a three-dimensional coordinate system, two lines are considered parallel if they never intersect and have the same direction vector (slope in 2D). ### Graphical Representation In the context of linear algebra and three-dimensional geometry: - Parallel lines imply that there is no intersection or point common between them. - For a system to have a unique solution, the equations (or geometrical representations such as planes or lines) must intersect at a single point. ### Conclusion Given the nature of parallel lines, they do not meet. Hence, a system of linear equations in three variables that includes two parallel lines cannot have a unique solution (1 solution) because the parallel lines inherently don't provide a single point of intersection. ### Answer - **False** ### Further Explanation In a system involving three dimensions (x, y, z), for a unique solution to exist, the three planes represented by the equations must intersect at exactly one point. Involving two parallel lines violates this condition, leading to either no solution or infinitely many solutions if the entire third plane aligns with one of the existing lines. Understanding these geometric relationships helps us interpret the behavior of equations and their solutions in higher dimensions, which is vital in linear algebra and its applications.
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