**Problem 2: Find the inverse of a matrix if it exists and use it to solve the system of linear equations. Be sure to show and explain your work.**Given system of linear equations:\[-2y + 5x = -44\]\[x + 5y = 2\]### Solution:1. **Form the system in matrix form \(AX = B\):** The system of equations can be written in matrix form \(AX = B\), where: \[ A = \begin{pmatrix} 5 & -2 \\ 1 & 5 \\ \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ \end{pmatrix}, \quad B = \begin{pmatrix} -44 \\ 2 \\ \end{pmatrix} \]2. **Find the inverse of matrix \(A\):** The inverse \(A^{-1}\) of a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by: \[ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \\ \end{pmatrix} \] For the matrix \(A = \begin{pmatrix} 5 & -2 \\ 1 & 5 \\ \end{pmatrix}\): \[ ad - bc = (5 \cdot 5) - (-2 \cdot 1) = 25 + 2 = 27 \] So, the inverse is: \[ A^{-1} = \frac{1}{27} \begin{pmatrix} 5 & 2 \\ -1 & 5 \\ \end{pmatrix} = \begin{pmatrix} \frac{5}{27} & \frac{2}{27} \\ -\frac{1}{27} & \frac{5}{27} \\ \end{pmatrix} \]3. **Solve for \(X

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Answered: 2. Find the inverse of a matrix if it…

2. Find the inverse of a matrix if it exists and use it to solve this system of linear equations. Be sure to show and explain your work. -2y + 5x = -44 x + 5y = 2

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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**Problem 2: Find the inverse of a matrix if it exists and use it to solve the system of linear equations. Be sure to show and explain your work.**

Given system of linear equations:
\[
-2y + 5x = -44
\]
\[
x + 5y = 2
\]

### Solution:

1. **Form the system in matrix form \(AX = B\):**

The system of equations can be written in matrix form \(AX = B\), where:
\[
A = \begin{pmatrix}
5 & -2 \\
1 & 5 \\
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y \\
\end{pmatrix}, \quad
B = \begin{pmatrix}
-44 \\
2 \\
\end{pmatrix}
\]

2. **Find the inverse of matrix \(A\):**

The inverse \(A^{-1}\) of a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by:
\[
A^{-1} = \frac{1}{ad - bc}
\begin{pmatrix}
d & -b \\
-c & a \\
\end{pmatrix}
\]
For the matrix \(A = \begin{pmatrix}
5 & -2 \\
1 & 5 \\
\end{pmatrix}\):
\[
ad - bc = (5 \cdot 5) - (-2 \cdot 1) = 25 + 2 = 27
\]
So, the inverse is:
\[
A^{-1} = \frac{1}{27}
\begin{pmatrix}
5 & 2 \\
-1 & 5 \\
\end{pmatrix}
= \begin{pmatrix}
\frac{5}{27} & \frac{2}{27} \\
-\frac{1}{27} & \frac{5}{27} \\
\end{pmatrix}
\]

3. **Solve for \(X

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