Suppose the solution set of a certain system of linear equations can be described as x₁ = 7+ 5x3, x₂ = −4 − 5x3, with x3 free. Use vectors to describe this set as a line in R³. Geometrically, the solution set is a line through parallel to
Suppose the solution set of a certain system of linear equations can be described as x₁ = 7+ 5x3, x₂ = −4 − 5x3, with x3 free. Use vectors to describe this set as a line in R³. Geometrically, the solution set is a line through parallel to
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Linear Equation Solutions in Vector Form
Suppose the solution set of a certain system of linear equations can be described as:
\[ x_1 = 7 + 5x_3, \]
\[ x_2 = -4 - 5x_3, \]
with \( x_3 \) free. Use vectors to describe this set as a line in \(\mathbb{R}^3\).
---
Geometrically, the solution set is a line through \(\boxed{ \begin{bmatrix} 7 \\ -4 \\ 0 \end{bmatrix} }\) parallel to \(\boxed{ \begin{bmatrix} 5 \\ -5 \\ 1 \end{bmatrix} }\).
### Explanation
Each variable \( x_1 \), \( x_2 \), and \( x_3 \) represents a coordinate in \(\mathbb{R}^3\). The parameters set up the line by giving a point through which the line passes and a direction vector indicating the line's direction.
**Point through which the line passes:**
\[ \begin{bmatrix} 7 \\ -4 \\ 0 \end{bmatrix} \]
**Parallel direction vector:**
\[ \begin{bmatrix} 5 \\ -5 \\ 1 \end{bmatrix} \]
### Understanding the Graph
In the context of \(\mathbb{R}^3\):
- The line passes through the point \((7, -4, 0)\).
- It extends parallel to the direction vector \((5, -5, 1)\).
Thus, the line in vector form can be written as:
\[ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 7 \\ -4 \\ 0 \end{bmatrix} + t \begin{bmatrix} 5 \\ -5 \\ 1 \end{bmatrix}, \quad t \in \mathbb{R} \]
Where \(t\) is a real number representing a parameter spanning all real values, indicating that this line extends indefinitely in both directions in \(\mathbb{R}^3\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2Ff3de2922-cd4e-4d30-b3ba-4244d6c419ce%2Fc17aa0a_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Equation Solutions in Vector Form
Suppose the solution set of a certain system of linear equations can be described as:
\[ x_1 = 7 + 5x_3, \]
\[ x_2 = -4 - 5x_3, \]
with \( x_3 \) free. Use vectors to describe this set as a line in \(\mathbb{R}^3\).
---
Geometrically, the solution set is a line through \(\boxed{ \begin{bmatrix} 7 \\ -4 \\ 0 \end{bmatrix} }\) parallel to \(\boxed{ \begin{bmatrix} 5 \\ -5 \\ 1 \end{bmatrix} }\).
### Explanation
Each variable \( x_1 \), \( x_2 \), and \( x_3 \) represents a coordinate in \(\mathbb{R}^3\). The parameters set up the line by giving a point through which the line passes and a direction vector indicating the line's direction.
**Point through which the line passes:**
\[ \begin{bmatrix} 7 \\ -4 \\ 0 \end{bmatrix} \]
**Parallel direction vector:**
\[ \begin{bmatrix} 5 \\ -5 \\ 1 \end{bmatrix} \]
### Understanding the Graph
In the context of \(\mathbb{R}^3\):
- The line passes through the point \((7, -4, 0)\).
- It extends parallel to the direction vector \((5, -5, 1)\).
Thus, the line in vector form can be written as:
\[ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 7 \\ -4 \\ 0 \end{bmatrix} + t \begin{bmatrix} 5 \\ -5 \\ 1 \end{bmatrix}, \quad t \in \mathbb{R} \]
Where \(t\) is a real number representing a parameter spanning all real values, indicating that this line extends indefinitely in both directions in \(\mathbb{R}^3\).
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