Substituting and Simplifying

Oct 13, 2025 by magicalceylontours.com 101 views

Introduction

In mathematics, solving systems of equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will consider a specific system of equations and analyze its solutions. The system consists of two equations involving quadratic terms, and our goal is to determine the number of solutions that satisfy both equations.

The System of Equations

The given system of equations is:

\begin{array}{l} -\frac{1}{3} x^2 = -\frac{5}{6} + \frac{1}{3} y^2 \\ 5 y^2 = \frac{25}{2} - 5 x^2 \end{array}

Rearranging the Equations

To simplify the analysis, we can rearrange the equations to isolate the quadratic terms. Multiplying the first equation by 3, we get:

-x^2 = -\frac{5}{2} + y^2

Multiplying the second equation by 2, we get:

10 y^2 = 25 - 10 x^2

Simplifying the Equations

We can further simplify the equations by dividing both sides of the first equation by -1 and both sides of the second equation by 10. This gives us:

x^2 = \frac{5}{2} - y^2

y^2 = \frac{5}{2} - \frac{1}{2} x^2

Substituting and Simplifying

We can substitute the expression for $y^2$ from the second equation into the first equation. This gives us:

x^2 = \frac{5}{2} - \left(\frac{5}{2} - \frac{1}{2} x^2\right)

Simplifying the equation, we get:

x^2 = \frac{1}{2} x^2

Analyzing the Solutions

From the simplified equation, we can see that $x^2 = 0$ or $x^2 = \frac{1}{2}$ . However, if $x^2 = \frac{1}{2}$ , then $y^2 = \frac{5}{2} - \frac{1}{2} x^2 = \frac{5}{2} - \frac{1}{2} \cdot \frac{1}{2} = \frac{5}{2} - \frac{1}{4} = \frac{9}{4}$ . This implies that $y^2$ is positive, but $x^2$ is not. Therefore, the only possible solution is $x^2 = 0$ .

Conclusion

Based on our analysis, we can conclude that the system of equations has only one solution, which is $x^2 = 0$ and $y^2 = \frac{5}{2}$ . This solution satisfies both equations simultaneously.

Final Answer

The final answer is B. one.

Discussion

The system of equations considered in this article is a quadratic system, and its solutions can be found by analyzing the properties of quadratic equations. The key to solving this system is to recognize that the two equations are equivalent and can be simplified to a single equation. By doing so, we can determine the number of solutions that satisfy both equations simultaneously.

Mathematical Concepts

The mathematical concepts used in this article include:

Quadratic equations: These are polynomial equations of degree two, and they can be written in the form $ax^2 + bx + c = 0$ .
Systems of equations: These are sets of equations that involve multiple variables and can be solved simultaneously.
Substitution: This is a technique used to solve systems of equations by substituting the expression for one variable into another equation.
Simplification: This is a technique used to simplify equations by combining like terms and eliminating unnecessary variables.

Real-World Applications

The concepts used in this article have real-world applications in various fields, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Systems of equations are used to design and optimize complex systems, such as electrical circuits, mechanical systems, and computer networks.
Economics: Quadratic equations are used to model the behavior of economic systems, including supply and demand curves, and the behavior of financial markets.

Conclusion

In conclusion, the system of equations considered in this article has only one solution, which is $x^2 = 0$ and $y^2 = \frac{5}{2}$ . This solution satisfies both equations simultaneously, and it can be found by analyzing the properties of quadratic equations and using techniques such as substitution and simplification. The mathematical concepts used in this article have real-world applications in various fields, including physics, engineering, and economics.

Q: What is the system of equations?

A: The system of equations is a set of two equations involving quadratic terms, which are:

\begin{array}{l} -\frac{1}{3} x^2 = -\frac{5}{6} + \frac{1}{3} y^2 \\ 5 y^2 = \frac{25}{2} - 5 x^2 \end{array}

Q: How many solutions does the system have?

A: The system of equations has only one solution, which is $x^2 = 0$ and $y^2 = \frac{5}{2}$ .

Q: What is the key to solving this system?

A: The key to solving this system is to recognize that the two equations are equivalent and can be simplified to a single equation.

Q: What mathematical concepts are used in this article?

A: The mathematical concepts used in this article include:

Quadratic equations: These are polynomial equations of degree two, and they can be written in the form $ax^2 + bx + c = 0$ .
Systems of equations: These are sets of equations that involve multiple variables and can be solved simultaneously.
Substitution: This is a technique used to solve systems of equations by substituting the expression for one variable into another equation.
Simplification: This is a technique used to simplify equations by combining like terms and eliminating unnecessary variables.

Q: What are the real-world applications of the concepts used in this article?

A: The concepts used in this article have real-world applications in various fields, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Systems of equations are used to design and optimize complex systems, such as electrical circuits, mechanical systems, and computer networks.
Economics: Quadratic equations are used to model the behavior of economic systems, including supply and demand curves, and the behavior of financial markets.

Q: How can I apply the concepts used in this article to my own work?

A: You can apply the concepts used in this article to your own work by:

Identifying quadratic equations: Look for equations that involve quadratic terms and can be written in the form $ax^2 + bx + c = 0$ .
Solving systems of equations: Use techniques such as substitution and simplification to solve systems of equations.
Applying mathematical concepts to real-world problems: Use mathematical concepts to model and analyze real-world problems in fields such as physics, engineering, and economics.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

Not recognizing equivalent equations: Make sure to recognize when two equations are equivalent and can be simplified to a single equation.
Not using substitution correctly: Make sure to use substitution correctly by substituting the expression for one variable into another equation.
Not simplifying equations correctly: Make sure to simplify equations correctly by combining like terms and eliminating unnecessary variables.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by:

Working through examples: Work through examples of systems of equations to practice solving them.
Using online resources: Use online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha to practice solving systems of equations.
Seeking help from a tutor or teacher: Seek help from a tutor or teacher if you are struggling to solve systems of equations.