Chapter6: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 7PS
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Question
![### Solving a Set of Equations by Graphing
Here is a problem where we solve a set of equations by graphing:
#### Given Equations:
1. \( y^2 = 16 - x^2 \)
2. \( y = 4 \)
#### Multiple Choice Answers:
- \( \quad\ \ \) \((0, 4)\)
- \( \quad\ \ \) \((0, -4)\)
- \( \quad\ \ \) no solutions
- \( \quad\ \ \) \((4, 0)\)
#### Explanation:
1. **Equation Analysis:**
- The first equation \( y^2 = 16 - x^2 \) represents a circle centered at the origin (0,0) with a radius of 4, because it can be rewritten as \( x^2 + y^2 = 16 \).
- The second equation \( y = 4 \) represents a horizontal line that intersects the y-axis at \( y = 4 \).
2. **Graphical Representation:**
- **Circle:** The equation \( x^2 + y^2 = 16 \) will create a circle with a radius of 4 units.
- **Horizontal Line:** The equation \( y = 4 \) is a straight line parallel to the x-axis and passes through the point (0, 4).
3. **Graph Intersection:**
To find the intersection points, we substitute \( y = 4 \) into the circle's equation:
\[
4^2 = 16 - x^2 \implies 16 = 16 - x^2 \implies x^2 = 0 \implies x = 0
\]
This gives us the point (0,4).
#### Conclusion:
The intersection point of the circle and the horizontal line is (0,4).
Thus, the correct answer is:
- \( \quad\ \ \) \((0, 4)\)
This is how we solve this set of equations by graphing, and the verified solution is the point where the two graphs intersect.
---
This problem illustrates how to solve a set of equations by graphically interpreting them and finding their points of intersection, a key concept in algebra and geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93aed08e-792c-4bc8-b542-6d972f8bdf88%2F6a94c42a-12a0-4c58-ab74-32afa50a7cb9%2Fhcf7qed_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a Set of Equations by Graphing
Here is a problem where we solve a set of equations by graphing:
#### Given Equations:
1. \( y^2 = 16 - x^2 \)
2. \( y = 4 \)
#### Multiple Choice Answers:
- \( \quad\ \ \) \((0, 4)\)
- \( \quad\ \ \) \((0, -4)\)
- \( \quad\ \ \) no solutions
- \( \quad\ \ \) \((4, 0)\)
#### Explanation:
1. **Equation Analysis:**
- The first equation \( y^2 = 16 - x^2 \) represents a circle centered at the origin (0,0) with a radius of 4, because it can be rewritten as \( x^2 + y^2 = 16 \).
- The second equation \( y = 4 \) represents a horizontal line that intersects the y-axis at \( y = 4 \).
2. **Graphical Representation:**
- **Circle:** The equation \( x^2 + y^2 = 16 \) will create a circle with a radius of 4 units.
- **Horizontal Line:** The equation \( y = 4 \) is a straight line parallel to the x-axis and passes through the point (0, 4).
3. **Graph Intersection:**
To find the intersection points, we substitute \( y = 4 \) into the circle's equation:
\[
4^2 = 16 - x^2 \implies 16 = 16 - x^2 \implies x^2 = 0 \implies x = 0
\]
This gives us the point (0,4).
#### Conclusion:
The intersection point of the circle and the horizontal line is (0,4).
Thus, the correct answer is:
- \( \quad\ \ \) \((0, 4)\)
This is how we solve this set of equations by graphing, and the verified solution is the point where the two graphs intersect.
---
This problem illustrates how to solve a set of equations by graphically interpreting them and finding their points of intersection, a key concept in algebra and geometry.
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