Use Cramer's rule to solve the system. 5x+8y=3 3x + 2y = 13 What is the solution of the system? X = y= (Type integers or simplified fractions.)
Use Cramer's rule to solve the system. 5x+8y=3 3x + 2y = 13 What is the solution of the system? X = y= (Type integers or simplified fractions.)
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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![## Section Title: Solving Systems of Equations Using Cramer's Rule
### Example Problem
**Given System of Equations:**
\[
5x + 8y = 3
\]
\[
3x + 2y = 13
\]
**Task:** Use Cramer's rule to solve the system.
### Solution Process
To solve this system using Cramer's rule, follow these steps:
1. **Write the System in Matrix Form:**
Let the system be represented as \(AX = B\), where:
\[
A = \begin{pmatrix}
5 & 8 \\
3 & 2 \\
\end{pmatrix},
X = \begin{pmatrix}
x \\
y \\
\end{pmatrix},
B = \begin{pmatrix}
3 \\
13 \\
\end{pmatrix}
\]
2. **Calculate Determinant of A (\(\Delta\)):**
\[
\Delta = \text{det}(A) = \begin{vmatrix}
5 & 8 \\
3 & 2 \\
\end{vmatrix} = 5(2) - 8(3) = 10 - 24 = -14
\]
3. **Calculate \(\Delta_{x}\):**
Replace the first column of \(A\) with \(B\):
\[
A_{x} = \begin{pmatrix}
3 & 8 \\
13 & 2 \\
\end{pmatrix}
\]
\[
\Delta_{x} = \text{det}(A_{x}) = \begin{vmatrix}
3 & 8 \\
13 & 2 \\
\end{vmatrix} = 3(2) - 8(13) = 6 - 104 = -98
\]
4. **Calculate \(\Delta_{y}\):**
Replace the second column of \(A\) with \(B\):
\[
A_{y} = \begin{pmatrix}
5 & 3 \\
3 & 13 \\
\end{pmatrix}
\]
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Transcribed Image Text:## Section Title: Solving Systems of Equations Using Cramer's Rule
### Example Problem
**Given System of Equations:**
\[
5x + 8y = 3
\]
\[
3x + 2y = 13
\]
**Task:** Use Cramer's rule to solve the system.
### Solution Process
To solve this system using Cramer's rule, follow these steps:
1. **Write the System in Matrix Form:**
Let the system be represented as \(AX = B\), where:
\[
A = \begin{pmatrix}
5 & 8 \\
3 & 2 \\
\end{pmatrix},
X = \begin{pmatrix}
x \\
y \\
\end{pmatrix},
B = \begin{pmatrix}
3 \\
13 \\
\end{pmatrix}
\]
2. **Calculate Determinant of A (\(\Delta\)):**
\[
\Delta = \text{det}(A) = \begin{vmatrix}
5 & 8 \\
3 & 2 \\
\end{vmatrix} = 5(2) - 8(3) = 10 - 24 = -14
\]
3. **Calculate \(\Delta_{x}\):**
Replace the first column of \(A\) with \(B\):
\[
A_{x} = \begin{pmatrix}
3 & 8 \\
13 & 2 \\
\end{pmatrix}
\]
\[
\Delta_{x} = \text{det}(A_{x}) = \begin{vmatrix}
3 & 8 \\
13 & 2 \\
\end{vmatrix} = 3(2) - 8(13) = 6 - 104 = -98
\]
4. **Calculate \(\Delta_{y}\):**
Replace the second column of \(A\) with \(B\):
\[
A_{y} = \begin{pmatrix}
5 & 3 \\
3 & 13 \\
\end{pmatrix}
\]
\
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