Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Show all work to verify if the given point is a solution to the system of equations.
![### System of Equations and Points
We have a system of two equations given by:
\[ 5x^2 - y^2 = 3 \]
\[ x^2 + 2y^2 = 5 \]
Additionally, we have a specific point mentioned:
\[ \text{Point:} \left( -1, -\sqrt{2} \right) \]
#### Explanation:
1. **Equation 1: \(5x^2 - y^2 = 3\)**
- This equation represents a conic section known as a hyperbola when both \(x\) and \(y\) are real numbers.
2. **Equation 2: \(x^2 + 2y^2 = 5\)**
- This equation represents an ellipse when both \(x\) and \(y\) are real numbers.
3. **Point: \(\left( -1, -\sqrt{2} \right)\)**
- This is a specific point in the Cartesian coordinate system which can be tested for its position relative to the conic sections defined by the above equations.
#### Graphical Representation:
To understand how these equations and points relate to each other, we would typically plot them on the Cartesian plane:
- **Hyperbola:**
- The hyperbola will have branches depending on the values of \(x\) and \(y\) that satisfy the first equation.
- **Ellipse:**
- The ellipse will have its shape determined by the second equation, showing an oval-like figure on the graph.
### Analysis
Verifying if the given point \(\left( -1, -\sqrt{2} \right)\) lies on either or both of the curves involves substituting \(x = -1\) and \(y = -\sqrt{2}\) into each of the equations.
1. **Substitute into Equation 1:**
\[5(-1)^2 - (-\sqrt{2})^2 = 5 - 2 = 3 \]
- Satisfies the first equation.
2. **Substitute into Equation 2:**
\[(-1)^2 + 2(-\sqrt{2})^2 = 1 + 4 = 5 \]
- Satisfies the second equation.
Since the point \(\left( -1, -\sqrt{2} \right)\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c9551-2207-4a1f-96cc-82c211723c9b%2Fa948e53e-c494-48ee-84fb-8c24ffddd796%2Fen61lpd_processed.png&w=3840&q=75)
Transcribed Image Text:### System of Equations and Points
We have a system of two equations given by:
\[ 5x^2 - y^2 = 3 \]
\[ x^2 + 2y^2 = 5 \]
Additionally, we have a specific point mentioned:
\[ \text{Point:} \left( -1, -\sqrt{2} \right) \]
#### Explanation:
1. **Equation 1: \(5x^2 - y^2 = 3\)**
- This equation represents a conic section known as a hyperbola when both \(x\) and \(y\) are real numbers.
2. **Equation 2: \(x^2 + 2y^2 = 5\)**
- This equation represents an ellipse when both \(x\) and \(y\) are real numbers.
3. **Point: \(\left( -1, -\sqrt{2} \right)\)**
- This is a specific point in the Cartesian coordinate system which can be tested for its position relative to the conic sections defined by the above equations.
#### Graphical Representation:
To understand how these equations and points relate to each other, we would typically plot them on the Cartesian plane:
- **Hyperbola:**
- The hyperbola will have branches depending on the values of \(x\) and \(y\) that satisfy the first equation.
- **Ellipse:**
- The ellipse will have its shape determined by the second equation, showing an oval-like figure on the graph.
### Analysis
Verifying if the given point \(\left( -1, -\sqrt{2} \right)\) lies on either or both of the curves involves substituting \(x = -1\) and \(y = -\sqrt{2}\) into each of the equations.
1. **Substitute into Equation 1:**
\[5(-1)^2 - (-\sqrt{2})^2 = 5 - 2 = 3 \]
- Satisfies the first equation.
2. **Substitute into Equation 2:**
\[(-1)^2 + 2(-\sqrt{2})^2 = 1 + 4 = 5 \]
- Satisfies the second equation.
Since the point \(\left( -1, -\sqrt{2} \right)\
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 11 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education