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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Question
last option is 2
![### Solving Systems of Nonlinear Equations
**Problem Statement:**
The following set of equations has how many solutions?
\[
\begin{align*}
y + x^2 &= 4 \\
y - x &= 4
\end{align*}
\]
**Options:**
- \(0\)
- Infinitely many
- \(1\)
**Solution Steps:**
1. **Substitute \(y\) from the second equation into the first equation:**
From the second equation:
\[
y = x + 4
\]
2. **Replace \(y\) in the first equation:**
\[
(x + 4) + x^2 = 4
\]
3. **Simplify the equation:**
\[
x^2 + x + 4 = 4
\]
4. **Subtract 4 from both sides:**
\[
x^2 + x = 0
\]
5. **Factor the equation:**
\[
x(x + 1) = 0
\]
6. **Solve for \(x\):**
\[
x = 0 \quad \text{or} \quad x = -1
\]
7. **Find the corresponding \(y\) values:**
For \(x = 0\):
\[
y = 0 + 4 = 4
\]
For \(x = -1\):
\[
y = -1 + 4 = 3
\]
**Conclusion:**
There are \(\boxed{2}\) solutions to this system of equations.
Thus, the correct answer to the original question is:
- \(1\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93aed08e-792c-4bc8-b542-6d972f8bdf88%2Ffd442145-33b6-4c72-9d4f-4f1982e6b04f%2Fujnshka_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Nonlinear Equations
**Problem Statement:**
The following set of equations has how many solutions?
\[
\begin{align*}
y + x^2 &= 4 \\
y - x &= 4
\end{align*}
\]
**Options:**
- \(0\)
- Infinitely many
- \(1\)
**Solution Steps:**
1. **Substitute \(y\) from the second equation into the first equation:**
From the second equation:
\[
y = x + 4
\]
2. **Replace \(y\) in the first equation:**
\[
(x + 4) + x^2 = 4
\]
3. **Simplify the equation:**
\[
x^2 + x + 4 = 4
\]
4. **Subtract 4 from both sides:**
\[
x^2 + x = 0
\]
5. **Factor the equation:**
\[
x(x + 1) = 0
\]
6. **Solve for \(x\):**
\[
x = 0 \quad \text{or} \quad x = -1
\]
7. **Find the corresponding \(y\) values:**
For \(x = 0\):
\[
y = 0 + 4 = 4
\]
For \(x = -1\):
\[
y = -1 + 4 = 3
\]
**Conclusion:**
There are \(\boxed{2}\) solutions to this system of equations.
Thus, the correct answer to the original question is:
- \(1\)
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