Find x, and x2. - 5 1 6- 15 -2 3 X2 - 3 17 X1 %3D X2 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![### Solving Systems of Linear Equations Using Matrices
**Objective:**
Solve for \( x_1 \) and \( x_2 \) given the matrix equation.
### Problem Statement:
Find \( x_1 \) and \( x_2 \).
\[
\begin{bmatrix}
-5 & 1 \\
-2 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
+
\begin{bmatrix}
-9 \\
-3
\end{bmatrix}
=
\begin{bmatrix}
15 \\
17
\end{bmatrix}
\]
### Solution:
We can break this problem into steps for clarity.
1. **Matrix Equation:**
The equation can be written as:
\[
\begin{bmatrix}
-5 & 1 \\
-2 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
+
\begin{bmatrix}
-9 \\
-3
\end{bmatrix}
=
\begin{bmatrix}
15 \\
17
\end{bmatrix}
\]
2. **Isolate the system of equations:**
To isolate the matrix containing \( x_1 \) and \( x_2 \), subtract \(\begin{bmatrix} -9 \\ -3 \end{bmatrix}\) from both sides:
\[
\begin{bmatrix}
-5 & 1 \\
-2 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
15 \\
17
\end{bmatrix}
-
\begin{bmatrix}
-9 \\
-3
\end{bmatrix}
\]
3. **Perform the subtraction:**
Simplify the right-hand side of the equation:
\[
\begin{bmatrix}
15 - (-9) \\
17 - (-3)
\end{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb3f5949-3440-4952-a068-9f8891699419%2F959a8b03-9f5a-4e85-a808-dd4cd9a426de%2Fww3o70v.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Linear Equations Using Matrices
**Objective:**
Solve for \( x_1 \) and \( x_2 \) given the matrix equation.
### Problem Statement:
Find \( x_1 \) and \( x_2 \).
\[
\begin{bmatrix}
-5 & 1 \\
-2 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
+
\begin{bmatrix}
-9 \\
-3
\end{bmatrix}
=
\begin{bmatrix}
15 \\
17
\end{bmatrix}
\]
### Solution:
We can break this problem into steps for clarity.
1. **Matrix Equation:**
The equation can be written as:
\[
\begin{bmatrix}
-5 & 1 \\
-2 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
+
\begin{bmatrix}
-9 \\
-3
\end{bmatrix}
=
\begin{bmatrix}
15 \\
17
\end{bmatrix}
\]
2. **Isolate the system of equations:**
To isolate the matrix containing \( x_1 \) and \( x_2 \), subtract \(\begin{bmatrix} -9 \\ -3 \end{bmatrix}\) from both sides:
\[
\begin{bmatrix}
-5 & 1 \\
-2 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
15 \\
17
\end{bmatrix}
-
\begin{bmatrix}
-9 \\
-3
\end{bmatrix}
\]
3. **Perform the subtraction:**
Simplify the right-hand side of the equation:
\[
\begin{bmatrix}
15 - (-9) \\
17 - (-3)
\end{
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