10x + y2 -4y = -13. Calculate and state the Graph the solution set to the equation: x defining features of the figure formed by the solution set [all work must be shown to receive credit].
10x + y2 -4y = -13. Calculate and state the Graph the solution set to the equation: x defining features of the figure formed by the solution set [all work must be shown to receive credit].
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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This is a geometry question.
![**Graphing the Solution Set to the Given Equation**
**Problem Statement:**
Graph the solution set to the equation: \( x^2 - 10x + y^2 - 4y = -13 \). Calculate and state the defining features of the figure formed by the solution set [all work must be shown to receive credit].
**Instructions:**
1. Rewrite the given equation in a form that makes it easy to recognize the figure it represents.
2. Complete the square for both \( x \) and \( y \) terms in the equation.
3. Graph the figure using the provided graph paper.
### Step-by-Step Solution:
1. **Starting Equation:**
\[ x^2 - 10x + y^2 - 4y = -13. \]
2. **Complete the Square:**
For \( x^2 - 10x \):
- Add and subtract \( (\frac{10}{2})^2 = 25 \):
\[ x^2 - 10x + 25 - 25 = x^2 - 10x + 25 - 25. \]
\[ (x - 5)^2 - 25. \]
For \( y^2 - 4y \):
- Add and subtract \( (\frac{4}{2})^2 = 4 \):
\[ y^2 - 4y + 4 - 4 = y^2 - 4y + 4 - 4. \]
\[ (y - 2)^2 - 4. \]
3. **Rewrite the equation using completed square terms:**
\[ (x - 5)^2 - 25 + (y - 2)^2 - 4 = -13. \]
4. **Simplify the equation:**
\[ (x - 5)^2 + (y - 2)^2 - 29 = -13. \]
\[ (x - 5)^2 + (y - 2)^2 = 16. \]
5. **Recognize the form of the equation:**
- The equation \((x - 5)^2 + (y - 2)^2 = 16\) represents a circle.
6. **Identify the defining features:**
- **Center**: The center of the circle is located at \((](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f58b3aa-f0aa-4022-88da-aea8e9637d1c%2Fe8ef3040-e5ba-4d7e-a993-7f533e2e325b%2F359pdmk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graphing the Solution Set to the Given Equation**
**Problem Statement:**
Graph the solution set to the equation: \( x^2 - 10x + y^2 - 4y = -13 \). Calculate and state the defining features of the figure formed by the solution set [all work must be shown to receive credit].
**Instructions:**
1. Rewrite the given equation in a form that makes it easy to recognize the figure it represents.
2. Complete the square for both \( x \) and \( y \) terms in the equation.
3. Graph the figure using the provided graph paper.
### Step-by-Step Solution:
1. **Starting Equation:**
\[ x^2 - 10x + y^2 - 4y = -13. \]
2. **Complete the Square:**
For \( x^2 - 10x \):
- Add and subtract \( (\frac{10}{2})^2 = 25 \):
\[ x^2 - 10x + 25 - 25 = x^2 - 10x + 25 - 25. \]
\[ (x - 5)^2 - 25. \]
For \( y^2 - 4y \):
- Add and subtract \( (\frac{4}{2})^2 = 4 \):
\[ y^2 - 4y + 4 - 4 = y^2 - 4y + 4 - 4. \]
\[ (y - 2)^2 - 4. \]
3. **Rewrite the equation using completed square terms:**
\[ (x - 5)^2 - 25 + (y - 2)^2 - 4 = -13. \]
4. **Simplify the equation:**
\[ (x - 5)^2 + (y - 2)^2 - 29 = -13. \]
\[ (x - 5)^2 + (y - 2)^2 = 16. \]
5. **Recognize the form of the equation:**
- The equation \((x - 5)^2 + (y - 2)^2 = 16\) represents a circle.
6. **Identify the defining features:**
- **Center**: The center of the circle is located at \((
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