Use Cramer's rule to compute the solution of the system. 6x₁ + 4x₂ + 2x3 = 6 2X1 + 3x3 = 4 - 2x1 + 4x2 = 2 x₁ = x₂ = x3 = ; ; (Type integers or simplified fractions.)
Use Cramer's rule to compute the solution of the system. 6x₁ + 4x₂ + 2x3 = 6 2X1 + 3x3 = 4 - 2x1 + 4x2 = 2 x₁ = x₂ = x3 = ; ; (Type integers or simplified fractions.)
Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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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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![### Solving a System of Linear Equations Using Cramer’s Rule
In this section, we will explore how to solve a system of linear equations using Cramer’s rule. Here is the given system of equations:
\[
6x_1 + 4x_2 + 2x_3 = 6
\]
\[
2x_1 + 3x_3 = 4
\]
\[
-2x_1 + 4x_2 = 2
\]
### Steps to Solve Using Cramer's Rule:
1. **Identify the Coefficient Matrix (A):**
The original system can be represented in matrix form \(A \mathbf{x} = \mathbf{b}\) where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column matrix of variables, and \(\mathbf{b}\) is the column matrix of constants on the right-hand side.
Coefficient Matrix \(A\):
\[
A = \begin{pmatrix}
6 & 4 & 2 \\
2 & 0 & 3 \\
-2 & 4 & 0
\end{pmatrix}
\]
2. **Determine the Determinant of Matrix A (\(\det(A)\)):**
\[
\det(A) = \begin{vmatrix}
6 & 4 & 2 \\
2 & 0 & 3 \\
-2 & 4 & 0
\end{vmatrix}
\]
(Compute this determinant using methods such as cofactor expansion.)
3. **Form the Matrices \(A_1, A_2, A_3\):**
Each matrix \(A_i\) is formed by replacing the i-th column of \(A\) with the matrix \(\mathbf{b}\):
\[
\mathbf{b} = \begin{pmatrix}
6 \\
4 \\
2
\end{pmatrix}
\]
\(A_1\) (replacing the first column with \(\mathbf{b}\)):
\[
A_1 = \begin{pmatrix}
6 & 4 & 2 \\
4 & 0 & 3 \\
2 &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2Fb0d700ec-382e-4ec4-93e8-fcd412309057%2Fy2bnsc_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving a System of Linear Equations Using Cramer’s Rule
In this section, we will explore how to solve a system of linear equations using Cramer’s rule. Here is the given system of equations:
\[
6x_1 + 4x_2 + 2x_3 = 6
\]
\[
2x_1 + 3x_3 = 4
\]
\[
-2x_1 + 4x_2 = 2
\]
### Steps to Solve Using Cramer's Rule:
1. **Identify the Coefficient Matrix (A):**
The original system can be represented in matrix form \(A \mathbf{x} = \mathbf{b}\) where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column matrix of variables, and \(\mathbf{b}\) is the column matrix of constants on the right-hand side.
Coefficient Matrix \(A\):
\[
A = \begin{pmatrix}
6 & 4 & 2 \\
2 & 0 & 3 \\
-2 & 4 & 0
\end{pmatrix}
\]
2. **Determine the Determinant of Matrix A (\(\det(A)\)):**
\[
\det(A) = \begin{vmatrix}
6 & 4 & 2 \\
2 & 0 & 3 \\
-2 & 4 & 0
\end{vmatrix}
\]
(Compute this determinant using methods such as cofactor expansion.)
3. **Form the Matrices \(A_1, A_2, A_3\):**
Each matrix \(A_i\) is formed by replacing the i-th column of \(A\) with the matrix \(\mathbf{b}\):
\[
\mathbf{b} = \begin{pmatrix}
6 \\
4 \\
2
\end{pmatrix}
\]
\(A_1\) (replacing the first column with \(\mathbf{b}\)):
\[
A_1 = \begin{pmatrix}
6 & 4 & 2 \\
4 & 0 & 3 \\
2 &
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