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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![**Solving a System of Equations by Substitution**
In this educational module, we will solve the following system of equations using the method of substitution:
\[ x^2 = 16 - y^2 \]
\[ 10 = y \]
### Step-by-Step Solution:
1. **Substitution:**
- The second equation directly gives the value of \( y \):
\[ y = 10 \]
2. **Substitute \( y = 10 \) into the first equation:**
\[ x^2 = 16 - (10)^2 \]
3. **Simplify the equation:**
\[ x^2 = 16 - 100 \]
\[ x^2 = -84 \]
At this point, we find that:
\[ x^2 = -84 \]
Since \( x^2 \) represents a square of a real number, and a square cannot be negative in the real number system, this equation does not have any real solutions.
### Conclusion:
The system of equations does not have any real solutions. Therefore, the answer is:
\[ \varnothing \text{ (the null set)} \]
### Answer Options:
- \( (4, 0) \)
- \( (0, 4) \)
- \( \left(2 \sqrt{21}, 10\right) \)
- \( \varnothing \text{ (the null set)} \)
Since we have demonstrated that the system has no real solutions, the correct answer is:
\[ \text{Ø (the null set)} \]
This interactive approach helps students understand the process of solving systems of equations by substitution, emphasizing the importance of verifying whether solutions are real or complex.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb4c9257-c3a4-42ab-bf0e-0de9c4c5a001%2F561a287f-0a40-4188-a7ca-68c3ffcc47f9%2F22fyd0b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving a System of Equations by Substitution**
In this educational module, we will solve the following system of equations using the method of substitution:
\[ x^2 = 16 - y^2 \]
\[ 10 = y \]
### Step-by-Step Solution:
1. **Substitution:**
- The second equation directly gives the value of \( y \):
\[ y = 10 \]
2. **Substitute \( y = 10 \) into the first equation:**
\[ x^2 = 16 - (10)^2 \]
3. **Simplify the equation:**
\[ x^2 = 16 - 100 \]
\[ x^2 = -84 \]
At this point, we find that:
\[ x^2 = -84 \]
Since \( x^2 \) represents a square of a real number, and a square cannot be negative in the real number system, this equation does not have any real solutions.
### Conclusion:
The system of equations does not have any real solutions. Therefore, the answer is:
\[ \varnothing \text{ (the null set)} \]
### Answer Options:
- \( (4, 0) \)
- \( (0, 4) \)
- \( \left(2 \sqrt{21}, 10\right) \)
- \( \varnothing \text{ (the null set)} \)
Since we have demonstrated that the system has no real solutions, the correct answer is:
\[ \text{Ø (the null set)} \]
This interactive approach helps students understand the process of solving systems of equations by substitution, emphasizing the importance of verifying whether solutions are real or complex.
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