**Topic: Solving Systems of Linear Equations Using Matrices** **Objective:** Show that \((-1, 2, 3)\) is the solution of the following system by making use of the form \(AX = B\). **System of Equations:** \[ \begin{align*} 2x + 3y - z &= 1 \\ x + 4y - z &= 4 \\ 3x + y + 2z &= 5 \\ \end{align*} \] **Explanation:** You are given a system of three linear equations with three variables \(x\), \(y\), and \(z\). The goal is to determine if the vector \((-1, 2, 3)\) is a solution to this system using the matrix equation \(AX = B\). - The matrix \(A\) represents the coefficients of the variables in the equations. - The matrix \(X\) represents the variables \(x\), \(y\), and \(z\). - The matrix \(B\) represents the constants on the right side of the equations. **Step-by-Step Approach:** 1. **Matrix Representation:** - Matrix \(A\): \[ A = \begin{bmatrix} 2 & 3 & -1 \\ 1 & 4 & -1 \\ 3 & 1 & 2 \\ \end{bmatrix} \] - Matrix \(X\): \[ X = \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} \] - Matrix \(B\): \[ B = \begin{bmatrix} 1 \\ 4 \\ 5 \\ \end{bmatrix} \] 2. **Substitute the proposed solution** \(X = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}\) into the matrix equation \(AX = B\) and verify if both sides are equal to confirm the solution. Use the matrix multiplication \(AX\) to verify if it equals \(B\). If the calculated product matches matrix \(B\), then \((-1, 2, 3)\) is indeed the solution to the system. **Conclusion:** The
**Topic: Solving Systems of Linear Equations Using Matrices** **Objective:** Show that \((-1, 2, 3)\) is the solution of the following system by making use of the form \(AX = B\). **System of Equations:** \[ \begin{align*} 2x + 3y - z &= 1 \\ x + 4y - z &= 4 \\ 3x + y + 2z &= 5 \\ \end{align*} \] **Explanation:** You are given a system of three linear equations with three variables \(x\), \(y\), and \(z\). The goal is to determine if the vector \((-1, 2, 3)\) is a solution to this system using the matrix equation \(AX = B\). - The matrix \(A\) represents the coefficients of the variables in the equations. - The matrix \(X\) represents the variables \(x\), \(y\), and \(z\). - The matrix \(B\) represents the constants on the right side of the equations. **Step-by-Step Approach:** 1. **Matrix Representation:** - Matrix \(A\): \[ A = \begin{bmatrix} 2 & 3 & -1 \\ 1 & 4 & -1 \\ 3 & 1 & 2 \\ \end{bmatrix} \] - Matrix \(X\): \[ X = \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} \] - Matrix \(B\): \[ B = \begin{bmatrix} 1 \\ 4 \\ 5 \\ \end{bmatrix} \] 2. **Substitute the proposed solution** \(X = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}\) into the matrix equation \(AX = B\) and verify if both sides are equal to confirm the solution. Use the matrix multiplication \(AX\) to verify if it equals \(B\). If the calculated product matches matrix \(B\), then \((-1, 2, 3)\) is indeed the solution to the system. **Conclusion:** The
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Topic: Solving Systems of Linear Equations Using Matrices**
**Objective:** Show that \((-1, 2, 3)\) is the solution of the following system by making use of the form \(AX = B\).
**System of Equations:**
\[
\begin{align*}
2x + 3y - z &= 1 \\
x + 4y - z &= 4 \\
3x + y + 2z &= 5 \\
\end{align*}
\]
**Explanation:**
You are given a system of three linear equations with three variables \(x\), \(y\), and \(z\). The goal is to determine if the vector \((-1, 2, 3)\) is a solution to this system using the matrix equation \(AX = B\).
- The matrix \(A\) represents the coefficients of the variables in the equations.
- The matrix \(X\) represents the variables \(x\), \(y\), and \(z\).
- The matrix \(B\) represents the constants on the right side of the equations.
**Step-by-Step Approach:**
1. **Matrix Representation:**
- Matrix \(A\):
\[
A = \begin{bmatrix}
2 & 3 & -1 \\
1 & 4 & -1 \\
3 & 1 & 2 \\
\end{bmatrix}
\]
- Matrix \(X\):
\[
X = \begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}
\]
- Matrix \(B\):
\[
B = \begin{bmatrix}
1 \\
4 \\
5 \\
\end{bmatrix}
\]
2. **Substitute the proposed solution** \(X = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}\) into the matrix equation \(AX = B\) and verify if both sides are equal to confirm the solution.
Use the matrix multiplication \(AX\) to verify if it equals \(B\). If the calculated product matches matrix \(B\), then \((-1, 2, 3)\) is indeed the solution to the system.
**Conclusion:**
The](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff76868b4-615a-41e7-bafb-d99979d90816%2Fa9315b6c-898e-4def-a4fa-b6d0e43d608b%2F8zjrp55.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Solving Systems of Linear Equations Using Matrices**
**Objective:** Show that \((-1, 2, 3)\) is the solution of the following system by making use of the form \(AX = B\).
**System of Equations:**
\[
\begin{align*}
2x + 3y - z &= 1 \\
x + 4y - z &= 4 \\
3x + y + 2z &= 5 \\
\end{align*}
\]
**Explanation:**
You are given a system of three linear equations with three variables \(x\), \(y\), and \(z\). The goal is to determine if the vector \((-1, 2, 3)\) is a solution to this system using the matrix equation \(AX = B\).
- The matrix \(A\) represents the coefficients of the variables in the equations.
- The matrix \(X\) represents the variables \(x\), \(y\), and \(z\).
- The matrix \(B\) represents the constants on the right side of the equations.
**Step-by-Step Approach:**
1. **Matrix Representation:**
- Matrix \(A\):
\[
A = \begin{bmatrix}
2 & 3 & -1 \\
1 & 4 & -1 \\
3 & 1 & 2 \\
\end{bmatrix}
\]
- Matrix \(X\):
\[
X = \begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}
\]
- Matrix \(B\):
\[
B = \begin{bmatrix}
1 \\
4 \\
5 \\
\end{bmatrix}
\]
2. **Substitute the proposed solution** \(X = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}\) into the matrix equation \(AX = B\) and verify if both sides are equal to confirm the solution.
Use the matrix multiplication \(AX\) to verify if it equals \(B\). If the calculated product matches matrix \(B\), then \((-1, 2, 3)\) is indeed the solution to the system.
**Conclusion:**
The
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