Let T: R2→R? be a linear transformation such that T(x1,x2) = (x1 +X2, 5x1 +6x2) . Find x such that T(x) = (3,10).
Let T: R2→R? be a linear transformation such that T(x1,x2) = (x1 +X2, 5x1 +6x2) . Find x such that T(x) = (3,10).
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
![Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation such that \( T(x_1, x_2) = (x_1 + x_2, 5x_1 + 6x_2) \). Find \(\mathbf{x}\) such that \( T(\mathbf{x}) = (3, 10) \).
**Solution:**
1. **Given:**
\[
T(x_1, x_2) = (x_1 + x_2, 5x_1 + 6x_2) = (3, 10)
\]
2. **Equations:**
\[
x_1 + x_2 = 3
\]
\[
5x_1 + 6x_2 = 10
\]
3. **Solve the System of Equations:**
- From the first equation:
\[
x_2 = 3 - x_1
\]
- Substitute \( x_2 = 3 - x_1 \) in the second equation:
\[
5x_1 + 6(3 - x_1) = 10
\]
\[
5x_1 + 18 - 6x_1 = 10
\]
\[
-x_1 + 18 = 10
\]
\[
-x_1 = -8
\]
\[
x_1 = 8
\]
- Substitute \( x_1 = 8 \) back into \( x_2 = 3 - x_1 \):
\[
x_2 = 3 - 8 = -5
\]
4. **Solution:**
\[
\mathbf{x} = (8, -5)
\]
Thus, \( x = (8, -5) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d31c5b4-c6dc-4739-be04-6ff777f4b60f%2Fd9f8d368-b363-4efd-8407-cec6e699bf47%2Flxywxpf_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation such that \( T(x_1, x_2) = (x_1 + x_2, 5x_1 + 6x_2) \). Find \(\mathbf{x}\) such that \( T(\mathbf{x}) = (3, 10) \).
**Solution:**
1. **Given:**
\[
T(x_1, x_2) = (x_1 + x_2, 5x_1 + 6x_2) = (3, 10)
\]
2. **Equations:**
\[
x_1 + x_2 = 3
\]
\[
5x_1 + 6x_2 = 10
\]
3. **Solve the System of Equations:**
- From the first equation:
\[
x_2 = 3 - x_1
\]
- Substitute \( x_2 = 3 - x_1 \) in the second equation:
\[
5x_1 + 6(3 - x_1) = 10
\]
\[
5x_1 + 18 - 6x_1 = 10
\]
\[
-x_1 + 18 = 10
\]
\[
-x_1 = -8
\]
\[
x_1 = 8
\]
- Substitute \( x_1 = 8 \) back into \( x_2 = 3 - x_1 \):
\[
x_2 = 3 - 8 = -5
\]
4. **Solution:**
\[
\mathbf{x} = (8, -5)
\]
Thus, \( x = (8, -5) \).
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