Equation Solvable By System Of Equations 3x^5-5x^3+2x^2-10x+4 And 4x^4+6x^3-11

JU07/09/2025 08, 2025 by THE IDEN 79 views

Which Equation Can Be Solved by Using This System of Equations?

In the realm of mathematics, systems of equations serve as powerful tools for modeling and solving a wide array of problems. These systems often involve multiple equations with multiple variables, and finding the solutions requires identifying the values that satisfy all equations simultaneously. When confronted with a system of equations, a crucial initial step is to recognize the underlying equations that can be derived from the system. This article delves into the process of identifying an equation that can be solved using a given system of equations, providing a comprehensive explanation and illustrative examples.

Understanding Systems of Equations

A system of equations is a set of two or more equations that share the same variables. The goal of solving a system of equations is to find the values of the variables that make all the equations in the system true. There are several methods for solving systems of equations, including substitution, elimination, and graphing. The most appropriate method depends on the specific nature of the equations involved. For instance, linear systems, where equations represent straight lines, can be efficiently solved using elimination or substitution. Nonlinear systems, which involve curves or more complex relationships, might require graphical methods or more advanced algebraic techniques.

When dealing with a system of equations, it's essential to understand the relationships between the equations. Often, one equation can be derived from the others by performing algebraic manipulations. Recognizing these derived equations can provide valuable insights into the system's solutions. For example, if we have two equations and can manipulate them to isolate a common expression, we can set those expressions equal to each other, creating a new equation that eliminates one variable. This simplification is a cornerstone of many solution strategies.

Identifying Equations Solvable by the System

To identify an equation that can be solved using a given system of equations, we need to look for relationships between the equations in the system. A common approach is to manipulate the equations algebraically to eliminate one or more variables. This process often involves substitution or elimination, where we rearrange one equation to isolate a variable and then substitute that expression into another equation. The resulting equation will have fewer variables, making it potentially easier to solve.

Consider the system of equations provided:

\left\{
\begin{array}{l}
y=3 x^5-5 x^3+2 x^2-10 x+4 \\
y=4 x^4+6 x^3-11
\end{array}
\right.

This system consists of two equations, both expressing y in terms of x. The first equation is a quintic polynomial, while the second is a quartic polynomial. To find an equation that can be solved using this system, we can equate the expressions for y from both equations. This is a valid step because, at the points where the solutions to the system exist, the y-values must be equal. By setting the two expressions equal, we eliminate y and obtain an equation involving only x.

Step-by-Step Solution

Equate the expressions for y:

Since both equations give an expression for y, we can set them equal to each other:
```
3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11
```
Rearrange the equation:

Now, we rearrange the equation to bring all terms to one side, setting the equation equal to zero:
```
3x^5 - 4x^4 - 5x^3 - 6x^3 + 2x^2 - 10x + 4 + 11 = 0
```
Combine like terms:
```
3x^5 - 4x^4 - 11x^3 + 2x^2 - 10x + 15 = 0
```

The resulting equation is a quintic polynomial equation in x. This equation can be solved to find the values of x that satisfy both equations in the original system. Solving this quintic equation can be challenging and may require numerical methods or specialized software, but it represents a single equation derived from the system that captures the relationship between x values at the intersection points of the two original equations.

Analysis of the Derived Equation

The equation we derived, 3x^5 - 4x^4 - 11x^3 + 2x^2 - 10x + 15 = 0, is a quintic polynomial equation. Quintic equations, in general, do not have a simple algebraic solution like quadratic equations. They often require numerical methods, such as the Newton-Raphson method or other iterative techniques, to find approximate solutions. The roots of this equation represent the x-coordinates of the points where the graphs of the two original equations intersect.

Each real root of the quintic equation corresponds to a solution of the system of equations. To find the corresponding y-values, we can substitute each real root back into either of the original equations. This will give us the y-coordinate for each intersection point. Complex roots of the quintic equation do not correspond to real intersection points in the Cartesian plane, but they are still mathematically valid solutions in the complex number system.

Comparison with Given Options

Now, let's compare the derived equation with the given options:

A. $3x^5 - 5x^3 + 2x^2 - 10x + 4 = 0$ B. $3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11$

Option A represents the first equation in the system set equal to zero. This equation, by itself, does not incorporate the information from the second equation. Therefore, solving it would only give the roots of the first equation, not the solutions to the system.

Option B, on the other hand, is precisely the equation we derived by equating the two expressions for y. This equation represents the condition where the y-values of both equations are equal, which is the foundation for finding solutions to the system. Therefore, Option B is the correct answer.

Practical Applications and Implications

Understanding how to derive and solve equations from systems has significant practical applications in various fields. In engineering, systems of equations are used to model complex systems and design components that meet specific criteria. In physics, they help describe the motion of objects and the interactions between forces. In economics, they are used to analyze market behavior and predict economic trends.

Solving systems of equations allows us to find the points where different mathematical relationships intersect. These intersection points often represent critical values or optimal solutions in real-world scenarios. For example, in a supply and demand model, the intersection point represents the equilibrium price and quantity. In optimization problems, the solutions to a system of equations may represent the points where a function reaches its maximum or minimum value.

Conclusion

In conclusion, the equation that can be solved by using the given system of equations is:

3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11

This equation is derived by equating the expressions for y from both equations in the system. Solving this equation will yield the x-values that satisfy both equations simultaneously. Understanding how to manipulate and derive equations from systems is a crucial skill in mathematics and its applications. This process allows us to simplify complex problems and find solutions that have practical significance in various fields. The ability to identify and solve these equations underscores the fundamental role of systems of equations in mathematical problem-solving and real-world applications. By mastering these techniques, one can approach complex mathematical challenges with greater confidence and insight. The journey through systems of equations not only enhances mathematical proficiency but also fosters a deeper appreciation for the interconnectedness of mathematical concepts and their relevance to the world around us. As we continue to explore the vast landscape of mathematics, the ability to solve systems of equations will undoubtedly remain a valuable asset, empowering us to tackle increasingly complex problems and make meaningful contributions to various fields of study and practice.