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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![## Solving Linear Systems
### Problem 4: Solve the following linear systems.
#### (a)
\[
\begin{cases}
x_1 + 2x_2 - x_3 = 3 \\
2x_1 + x_2 + x_4 = 4 \\
x_1 - x_2 + x_3 + x_4 = 1
\end{cases}
\]
#### (b)
\[
\begin{cases}
x + 2x_2 + 3x_3 + x_4 - x_5 = 1 \\
3x_1 - x_2 + x_3 + x_4 + x_5 = 3
\end{cases}
\]
### Explanation
**Part (a)** consists of a system of three linear equations with four unknowns: \(x_1, x_2, x_3, x_4\). Each equation represents a linear combination of these variables set equal to a constant.
**Part (b)** features two linear equations with five unknowns: \(x, x_2, x_3, x_4, x_5\). These equations also represent linear combinations of the unknowns, each equated to a constant.
To solve these systems, techniques such as substitution, elimination, or matrix methods (like Gaussian elimination) can be utilized to find the values of the unknown variables that satisfy all equations simultaneously.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3574180-5be7-46af-9f18-c1fbb9ec682c%2Ff15af15a-6af1-48c0-85e7-6364699b2c42%2Fe3v78t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Solving Linear Systems
### Problem 4: Solve the following linear systems.
#### (a)
\[
\begin{cases}
x_1 + 2x_2 - x_3 = 3 \\
2x_1 + x_2 + x_4 = 4 \\
x_1 - x_2 + x_3 + x_4 = 1
\end{cases}
\]
#### (b)
\[
\begin{cases}
x + 2x_2 + 3x_3 + x_4 - x_5 = 1 \\
3x_1 - x_2 + x_3 + x_4 + x_5 = 3
\end{cases}
\]
### Explanation
**Part (a)** consists of a system of three linear equations with four unknowns: \(x_1, x_2, x_3, x_4\). Each equation represents a linear combination of these variables set equal to a constant.
**Part (b)** features two linear equations with five unknowns: \(x, x_2, x_3, x_4, x_5\). These equations also represent linear combinations of the unknowns, each equated to a constant.
To solve these systems, techniques such as substitution, elimination, or matrix methods (like Gaussian elimination) can be utilized to find the values of the unknown variables that satisfy all equations simultaneously.
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