Find the solution set by finding the inverse of the coefficient matrix and using the inverse to solve the system. / 2х — у %3D 3 (4х + Зу 3D 1

Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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**Title: Solving Systems of Linear Equations Using Matrix Inversion**

To find the solution set for the given system of linear equations, we will utilize the inverse of the coefficient matrix.

### System of Equations:

\[
\begin{align*}
2x - y &= 3 \\
4x + 3y &= 1 \\
\end{align*}
\]

**Steps to Solve:**

1. **Formulate the Coefficient Matrix:**

   The coefficient matrix \( A \) is derived from the coefficients of \( x \) and \( y \) in the system of equations:
   \[
   A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}
   \]

2. **Formulate the Constant Matrix:**

   The constant matrix \( B \) from the right-hand side of the equations is:
   \[
   B = \begin{bmatrix} 3 \\ 1 \end{bmatrix}
   \]

3. **Compute the Inverse of Matrix \( A \):**

   To find the inverse of matrix \( A \), we follow the standard procedure for finding a 2x2 matrix inverse. If \( A^{-1} \) exists, it is given by:
   \[
   A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
   \]
   where \( a, b, c, \) and \( d \) are elements in \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).

4. **Determine the Solution Matrix \( X \):**

   Once the inverse matrix \( A^{-1} \) is computed, the solution matrix \( X \) is found by multiplying \( A^{-1} \) by the constant matrix \( B \):
   \[
   X = A^{-1}B
   \]

5. **Verify the Solution:**

   It's essential to verify the computed solution by substituting back into the original equations.

This approach provides a robust method to find the solution for a system of linear equations using matrix techniques, specifically matrix inversion.
Transcribed Image Text:**Title: Solving Systems of Linear Equations Using Matrix Inversion** To find the solution set for the given system of linear equations, we will utilize the inverse of the coefficient matrix. ### System of Equations: \[ \begin{align*} 2x - y &= 3 \\ 4x + 3y &= 1 \\ \end{align*} \] **Steps to Solve:** 1. **Formulate the Coefficient Matrix:** The coefficient matrix \( A \) is derived from the coefficients of \( x \) and \( y \) in the system of equations: \[ A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix} \] 2. **Formulate the Constant Matrix:** The constant matrix \( B \) from the right-hand side of the equations is: \[ B = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \] 3. **Compute the Inverse of Matrix \( A \):** To find the inverse of matrix \( A \), we follow the standard procedure for finding a 2x2 matrix inverse. If \( A^{-1} \) exists, it is given by: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] where \( a, b, c, \) and \( d \) are elements in \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). 4. **Determine the Solution Matrix \( X \):** Once the inverse matrix \( A^{-1} \) is computed, the solution matrix \( X \) is found by multiplying \( A^{-1} \) by the constant matrix \( B \): \[ X = A^{-1}B \] 5. **Verify the Solution:** It's essential to verify the computed solution by substituting back into the original equations. This approach provides a robust method to find the solution for a system of linear equations using matrix techniques, specifically matrix inversion.
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