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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Show full answers and steps to part b) & c)
![Sure, here's a detailed transcription suitable for an educational website:
---
### Linear Systems and Gaussian Elimination
**Objective:** Utilize Elementary Row Reduction and Gaussian elimination to solve the following linear systems:
#### (a)
\[
\begin{cases}
2x_1 + x_2 = 6 \\
3x_1 + 5x_2 = 2
\end{cases}
\]
This system of equations can be solved using Elementary Row Operations to achieve a triangular matrix and subsequently using back substitution.
#### (b)
\[
\begin{cases}
2x_1 + 2x_2 + 2x_3 = 0 \\
-2x_1 + 5x_2 + 3x_3 = 0 \\
-2x_1 + 7x_2 + x_3 = 0
\end{cases}
\]
In this system, we have three equations with three variables. Applying Gaussian elimination will simplify the matrix to determine the values of \(x_1\), \(x_2\), and \(x_3\).
#### (c)
\[
\begin{cases}
3x_1 - 2x_2 + 10x_3 = -8 \\
4x_1 + x_3 = -6 \\
4x_3 = -2
\end{cases}
\]
For this system, a combination of direct substitution and Gaussian elimination will allow us to solve for the unknowns.
### Explanation of Steps:
1. **Convert the system of equations into an augmented matrix.**
2. **Apply Elementary Row Operations** (swap rows, multiply a row by a non-zero scalar, add or subtract the scalar multiple of one row to another) to simplify the matrix to row-echelon form or reduced row-echelon form.
3. **Solve the resulting triangular matrix** for the variables using back substitution or direct calculation if necessary.
These exercises aim to enhance understanding of solving systems of linear equations efficiently using matrix operations.
---
This transcription provides a clear guide for educators and students on the methodology of solving linear systems using Gaussian elimination.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3574180-5be7-46af-9f18-c1fbb9ec682c%2Fd761b8da-7d4e-4b72-bc41-9447a765619c%2Fxh4x50s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sure, here's a detailed transcription suitable for an educational website:
---
### Linear Systems and Gaussian Elimination
**Objective:** Utilize Elementary Row Reduction and Gaussian elimination to solve the following linear systems:
#### (a)
\[
\begin{cases}
2x_1 + x_2 = 6 \\
3x_1 + 5x_2 = 2
\end{cases}
\]
This system of equations can be solved using Elementary Row Operations to achieve a triangular matrix and subsequently using back substitution.
#### (b)
\[
\begin{cases}
2x_1 + 2x_2 + 2x_3 = 0 \\
-2x_1 + 5x_2 + 3x_3 = 0 \\
-2x_1 + 7x_2 + x_3 = 0
\end{cases}
\]
In this system, we have three equations with three variables. Applying Gaussian elimination will simplify the matrix to determine the values of \(x_1\), \(x_2\), and \(x_3\).
#### (c)
\[
\begin{cases}
3x_1 - 2x_2 + 10x_3 = -8 \\
4x_1 + x_3 = -6 \\
4x_3 = -2
\end{cases}
\]
For this system, a combination of direct substitution and Gaussian elimination will allow us to solve for the unknowns.
### Explanation of Steps:
1. **Convert the system of equations into an augmented matrix.**
2. **Apply Elementary Row Operations** (swap rows, multiply a row by a non-zero scalar, add or subtract the scalar multiple of one row to another) to simplify the matrix to row-echelon form or reduced row-echelon form.
3. **Solve the resulting triangular matrix** for the variables using back substitution or direct calculation if necessary.
These exercises aim to enhance understanding of solving systems of linear equations efficiently using matrix operations.
---
This transcription provides a clear guide for educators and students on the methodology of solving linear systems using Gaussian elimination.
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