How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there for which Preview
How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there for which Preview
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement**
Determine the number of integer solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 = 31 \]
subject to the constraints:
\[ x_1 \geq 4, \, x_2 \geq 2, \, x_3 \geq 5, \, \text{and} \, x_4 \geq 3. \]
**Solution Approach**
To find the number of integer solutions, we can use a transformation such as \( y_1 = x_1 - 4 \), \( y_2 = x_2 - 2 \), \( y_3 = x_3 - 5 \), and \( y_4 = x_4 - 3 \) to simplify the constraints \( y_1, y_2, y_3, y_4 \geq 0 \).
Substituting into the equation:
\[ (y_1 + 4) + (y_2 + 2) + (y_3 + 5) + (y_4 + 3) = 31 \]
Simplifies to:
\[ y_1 + y_2 + y_3 + y_4 = 17 \]
Now, you can find the number of non-negative integer solutions to this equation using methods like generating functions or combinatorial techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F140c7626-2888-4935-8928-dcb55a4fc57b%2F6377c3dc-c5a5-47b3-9ba3-ccd462d95b63%2Fps3vmgd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Determine the number of integer solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 = 31 \]
subject to the constraints:
\[ x_1 \geq 4, \, x_2 \geq 2, \, x_3 \geq 5, \, \text{and} \, x_4 \geq 3. \]
**Solution Approach**
To find the number of integer solutions, we can use a transformation such as \( y_1 = x_1 - 4 \), \( y_2 = x_2 - 2 \), \( y_3 = x_3 - 5 \), and \( y_4 = x_4 - 3 \) to simplify the constraints \( y_1, y_2, y_3, y_4 \geq 0 \).
Substituting into the equation:
\[ (y_1 + 4) + (y_2 + 2) + (y_3 + 5) + (y_4 + 3) = 31 \]
Simplifies to:
\[ y_1 + y_2 + y_3 + y_4 = 17 \]
Now, you can find the number of non-negative integer solutions to this equation using methods like generating functions or combinatorial techniques.
Expert Solution
Step 1
Find change all inequality to grater then equal to 0. Then use the theorem
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
