Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Use Cramer's Rule to Solve
![**Problem 2: Solving Simultaneous Equations Using Cramer's Rule**
Use Cramer's Rule to solve the following system of linear equations:
\[
\begin{align*}
x + 7y &= 23 \\
6x + y &= 15
\end{align*}
\]
**Cramer's Rule Overview:**
Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.
To use Cramer's Rule, follow these steps:
1. **Determine the Coefficient Matrix (A):**
\[
A = \begin{bmatrix}
1 & 7 \\
6 & 1
\end{bmatrix}
\]
2. **Calculate the Determinant of A (\(|A|\)):**
\[
|A| = (1)(1) - (7)(6) = 1 - 42 = -41
\]
3. **Determine the Matrices for \(x\) and \(y\):**
For \(x\), replace the first column of A with the constants:
\[
A_x = \begin{bmatrix}
23 & 7 \\
15 & 1
\end{bmatrix}
\]
For \(y\), replace the second column of A with the constants:
\[
A_y = \begin{bmatrix}
1 & 23 \\
6 & 15
\end{bmatrix}
\]
4. **Calculate the Determinant of \(A_x\) and \(A_y\):**
\[
|A_x| = (23)(1) - (7)(15) = 23 - 105 = -82
\]
\[
|A_y| = (1)(15) - (23)(6) = 15 - 138 = -123
\]
5. **Apply Cramer's Rule to find \(x\) and \(y\):**
\[
x = \frac{|A_x|}{|A|} = \frac{-82}{-41} = 2
\]
\[
y = \frac{|A_y|}{|A|} = \frac{-123](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F681a1dd5-ea79-4df0-af09-f7e014c18f3f%2F1d357490-84b1-4700-8f9a-11dcbcfab0f2%2F2u01rm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Solving Simultaneous Equations Using Cramer's Rule**
Use Cramer's Rule to solve the following system of linear equations:
\[
\begin{align*}
x + 7y &= 23 \\
6x + y &= 15
\end{align*}
\]
**Cramer's Rule Overview:**
Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.
To use Cramer's Rule, follow these steps:
1. **Determine the Coefficient Matrix (A):**
\[
A = \begin{bmatrix}
1 & 7 \\
6 & 1
\end{bmatrix}
\]
2. **Calculate the Determinant of A (\(|A|\)):**
\[
|A| = (1)(1) - (7)(6) = 1 - 42 = -41
\]
3. **Determine the Matrices for \(x\) and \(y\):**
For \(x\), replace the first column of A with the constants:
\[
A_x = \begin{bmatrix}
23 & 7 \\
15 & 1
\end{bmatrix}
\]
For \(y\), replace the second column of A with the constants:
\[
A_y = \begin{bmatrix}
1 & 23 \\
6 & 15
\end{bmatrix}
\]
4. **Calculate the Determinant of \(A_x\) and \(A_y\):**
\[
|A_x| = (23)(1) - (7)(15) = 23 - 105 = -82
\]
\[
|A_y| = (1)(15) - (23)(6) = 15 - 138 = -123
\]
5. **Apply Cramer's Rule to find \(x\) and \(y\):**
\[
x = \frac{|A_x|}{|A|} = \frac{-82}{-41} = 2
\]
\[
y = \frac{|A_y|}{|A|} = \frac{-123
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