x, - 2x2 - x3 = 5 - 3x, - 6x2 - 2x3 = 2
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter7: Conic Sections And Quadratic Systems
Section7.4: Solving Nonlinear Systems Of Equations
Problem 63E
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![The image presented contains a system of linear equations, which is commonly encountered in linear algebra and related fields. The system of equations is as follows:
\[ x_1 - 2x_2 - x_3 = 5 \]
\[ 3x_1 - 6x_2 - 2x_3 = 2 \]
Each equation is a linear relationship between the variables \(x_1\), \(x_2\), and \(x_3\). This system can be solved using various methods such as substitution, elimination, or matrix operations.
**Explanation of Terms:**
- \(x_1, x_2, x_3\): Variables that the equations are solving for.
- \(= 5, = 2\): Constants on the right-hand side of the equations.
To understand and solve these equations, you might perform the following steps:
- **Substitution:** Solve one equation for one variable in terms of the others and substitute into the other equation(s).
- **Elimination:** Multiply or add equations together to eliminate one variable, simplifying the system to fewer variables.
- **Matrix Operations:** Represent the system as a matrix and use techniques like row reduction to find the solution.
**Graphical Representation:**
Often, such systems can be visualized graphically, where each equation represents a plane in three-dimensional space. The solution to the system corresponds to the intersection of these planes. If the planes intersect at a single point, the solution is unique. If they intersect along a line or infinitely many points, the system has infinitely many solutions, or if they do not intersect, the system has no solution.
Understanding the behavior of such systems has practical applications in various real-world problems, including engineering, economics, and computer science.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92524007-eb79-4716-97da-6681fe12a787%2Fd6334b01-7d1c-44e4-8e3e-55e203d3bcc6%2F7ax7a3l_processed.png&w=3840&q=75)
Transcribed Image Text:The image presented contains a system of linear equations, which is commonly encountered in linear algebra and related fields. The system of equations is as follows:
\[ x_1 - 2x_2 - x_3 = 5 \]
\[ 3x_1 - 6x_2 - 2x_3 = 2 \]
Each equation is a linear relationship between the variables \(x_1\), \(x_2\), and \(x_3\). This system can be solved using various methods such as substitution, elimination, or matrix operations.
**Explanation of Terms:**
- \(x_1, x_2, x_3\): Variables that the equations are solving for.
- \(= 5, = 2\): Constants on the right-hand side of the equations.
To understand and solve these equations, you might perform the following steps:
- **Substitution:** Solve one equation for one variable in terms of the others and substitute into the other equation(s).
- **Elimination:** Multiply or add equations together to eliminate one variable, simplifying the system to fewer variables.
- **Matrix Operations:** Represent the system as a matrix and use techniques like row reduction to find the solution.
**Graphical Representation:**
Often, such systems can be visualized graphically, where each equation represents a plane in three-dimensional space. The solution to the system corresponds to the intersection of these planes. If the planes intersect at a single point, the solution is unique. If they intersect along a line or infinitely many points, the system has infinitely many solutions, or if they do not intersect, the system has no solution.
Understanding the behavior of such systems has practical applications in various real-world problems, including engineering, economics, and computer science.
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