Use the inverse of the coefficient matrix to solve the system of equations. x + 2y + 3z = 10 2x + 3y + 2z = 8 -x-2y-4z = 0 (x,y,z) = (Type an ordered triple, using integers or fractions.) ...
Use the inverse of the coefficient matrix to solve the system of equations. x + 2y + 3z = 10 2x + 3y + 2z = 8 -x-2y-4z = 0 (x,y,z) = (Type an ordered triple, using integers or 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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Question
![**Solving Systems of Equations Using the Inverse of a Coefficient Matrix**
To find the solution to the given system of equations, you can use the inverse of the coefficient matrix method. Here's the system of equations provided:
1. \( x + 2y + 3z = 10 \)
2. \( 2x + 3y + 2z = 8 \)
3. \( -x - 2y - 4z = 0 \)
**Steps to Solve:**
1. **Formulate the Coefficient Matrix (A):**
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
2 & 3 & 2 \\
-1 & -2 & -4
\end{bmatrix}
\]
2. **Formulate the Constant Matrix (B):**
\[
B = \begin{bmatrix}
10 \\
8 \\
0
\end{bmatrix}
\]
3. **Find the Inverse of the Coefficient Matrix (A⁻¹):**
Calculate \( A⁻¹ \) using standard methods like row operations or a calculator capable of matrix computations.
4. **Solve for the Variable Matrix (X):**
Use the formula \( X = A^{-1}B \). This will give you:
\[
X = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
\]
5. **Write the Solution as an Ordered Triple:**
You are required to enter the solution as an ordered triple \((x, y, z)\), using integers or fractions.
**Input Form:**
\[
(x, y, z) = \boxed{\phantom{42}}, \boxed{\phantom{42}}, \boxed{\phantom{42}}
\]
(Type an ordered triple, using integers or fractions.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09ace70e-38b4-4990-b7ba-acac1bf1dd80%2F97024283-326d-42ae-ac4c-3e1802031fde%2F7xqkcci_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Systems of Equations Using the Inverse of a Coefficient Matrix**
To find the solution to the given system of equations, you can use the inverse of the coefficient matrix method. Here's the system of equations provided:
1. \( x + 2y + 3z = 10 \)
2. \( 2x + 3y + 2z = 8 \)
3. \( -x - 2y - 4z = 0 \)
**Steps to Solve:**
1. **Formulate the Coefficient Matrix (A):**
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
2 & 3 & 2 \\
-1 & -2 & -4
\end{bmatrix}
\]
2. **Formulate the Constant Matrix (B):**
\[
B = \begin{bmatrix}
10 \\
8 \\
0
\end{bmatrix}
\]
3. **Find the Inverse of the Coefficient Matrix (A⁻¹):**
Calculate \( A⁻¹ \) using standard methods like row operations or a calculator capable of matrix computations.
4. **Solve for the Variable Matrix (X):**
Use the formula \( X = A^{-1}B \). This will give you:
\[
X = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
\]
5. **Write the Solution as an Ordered Triple:**
You are required to enter the solution as an ordered triple \((x, y, z)\), using integers or fractions.
**Input Form:**
\[
(x, y, z) = \boxed{\phantom{42}}, \boxed{\phantom{42}}, \boxed{\phantom{42}}
\]
(Type an ordered triple, using integers or fractions.)
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