Solving Equations With Systems Of Equations A Comprehensive Guide

In the realm of mathematics, systems of equations provide a powerful tool for solving complex problems. These systems, comprising two or more equations, can be manipulated to isolate variables and find solutions that satisfy all equations simultaneously. One common application of systems of equations is to solve a single equation by transforming it into a system. In this article, we will explore how a specific system of equations can be used to solve a given equation. This involves understanding the underlying principles of equation solving, recognizing the structure of the system, and applying algebraic techniques to arrive at the solution. By delving into this topic, we aim to enhance your understanding of mathematical problem-solving and equip you with the skills to tackle similar challenges. Our main focus will be on how the given system of equations, which involves polynomial functions, can be utilized to find the solution to a related equation. We will break down the steps, explain the logic behind each manipulation, and provide clear examples to illustrate the concepts. This approach will not only help you understand the specific problem at hand but also build a foundation for solving more complex mathematical problems in the future. The key to understanding this process lies in recognizing that the solutions to the system of equations are the points where the graphs of the equations intersect. This intersection represents the values of ${ x }$ and ${ y }$ that satisfy both equations, thus providing a solution to the original equation derived from the system. Throughout this discussion, we will emphasize the importance of careful algebraic manipulation and logical reasoning in arriving at the correct solution. By the end of this article, you should be able to confidently identify which equation can be solved using the given system and understand the broader applications of this method in mathematics.

Understanding the System of Equations

The provided system of equations is as follows:

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

This system consists of two equations, both expressing ${ y }$ in terms of ${ x }$. The first equation, ${ y = 3x^5 - 5x^3 + 2x^2 - 10x + 4 }$, is a quintic polynomial, which means it is a polynomial of degree five. The second equation, ${ y = 4x^4 + 6x^3 - 11 }$, is a quartic polynomial, a polynomial of degree four. To solve a system of equations like this, we aim to find the values of ${ x }$ and ${ y }$ that satisfy both equations simultaneously. This can be achieved through various methods, such as substitution, elimination, or graphical methods. In this particular case, the structure of the equations lends itself well to the substitution method. Since both equations are already expressed in the form ${ y = f(x) }$, we can set the expressions for ${ y }$ equal to each other. This creates a new equation in terms of ${ x }$ alone, which we can then solve to find the possible values of ${ x }$. Understanding the nature of these polynomial equations is crucial for solving the system. Polynomial equations can have multiple real and complex roots, and the degree of the polynomial indicates the maximum number of roots it can have. For instance, the quintic polynomial can have up to five roots, while the quartic polynomial can have up to four roots. When we equate the two expressions for ${ y }$, we will obtain a new polynomial equation whose roots correspond to the ${ x }$-values where the two original curves intersect. These ${ x }$-values are the solutions to the equation we are trying to find. The graphical interpretation of this system is also important. Each equation represents a curve in the ${ xy }$-plane. The solutions to the system correspond to the points where these curves intersect. Therefore, by finding the ${ x }$-coordinates of these intersection points, we can solve the equation derived from setting the two expressions for ${ y }$ equal to each other. This visual perspective can often provide additional insights and help in verifying the algebraic solutions.

Identifying the Solvable Equation

To determine which equation can be solved using the given system, we need to manipulate the system algebraically. As mentioned earlier, both equations are in the form ${ y = f(x) }$, making substitution an effective method. We can set the two expressions for ${ y }$ equal to each other:

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

This step is crucial because it eliminates ${ y }$ and gives us a single equation in terms of ${ x }$. This equation represents the ${ x }$-values where the two original equations intersect. Now, let's analyze the given options:

• A. 3x⁵ - 5x³ + 2x² - 10x + 4 = 0
• B. 3x⁵ - 5x³ + 2x² - 10x + 4 = 4x⁴ + 6x³ - 11

Comparing these options with the equation we derived, we can see that option B, 3x⁵ - 5x³ + 2x² - 10x + 4 = 4x⁴ + 6x³ - 11, directly matches the equation obtained by equating the two expressions for ${ y }$. Option A, 3x⁵ - 5x³ + 2x² - 10x + 4 = 0, is different. It represents the condition where the first equation, ${ y = 3x^5 - 5x^3 + 2x^2 - 10x + 4 }$, equals zero. This is not the equation we derived from the system. The equation we derived represents the condition where the two original equations are equal, i.e., where their graphs intersect. To solve option A, one would need to find the roots of the quintic polynomial 3x⁵ - 5x³ + 2x² - 10x + 4, which is a separate problem from solving the system of equations. Therefore, the correct equation that can be solved using the given system is option B. This approach highlights the importance of recognizing the relationship between systems of equations and single equations. By equating the expressions for ${ y }$, we transform the system into a single equation that captures the intersection points of the two original equations. This transformation is a powerful technique in algebra and is widely used in various mathematical applications. The process of identifying the correct equation also reinforces the understanding of what a solution to a system of equations represents. It is the set of values that satisfy all equations in the system simultaneously, which in this case, translates to the ${ x }$-values that make the two expressions for ${ y }$ equal.

Step-by-Step Solution and Explanation

To solve the equation 3x⁵ - 5x³ + 2x² - 10x + 4 = 4x⁴ + 6x³ - 11, which we identified as the equation that can be solved using the given system, we need to rearrange and simplify the equation. The initial equation is:

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

The first step is to move all terms to one side of the equation to set it equal to zero. This is a standard practice in solving polynomial equations, as it allows us to use methods like factoring, the rational root theorem, or numerical methods to find the solutions. Subtracting 4x⁴ + 6x³ - 11 from both sides, we get:

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

Combining like terms, we simplify the equation:

$$3x^5 - 4x^4 - 11x^3 + 2x^2 - 10x + 15 = 0$$

This is a quintic polynomial equation, which is generally difficult to solve analytically. There is no general algebraic formula for solving quintic equations, unlike quadratic, cubic, or quartic equations. Therefore, we often resort to numerical methods or graphical methods to approximate the solutions. Numerical methods, such as the Newton-Raphson method or bisection method, can be used to find approximate roots to a high degree of accuracy. These methods involve iterative calculations and are often implemented using computer software. Graphical methods involve plotting the polynomial function and identifying the points where the graph intersects the x-axis. These intersection points represent the real roots of the equation. Factoring, if possible, can simplify the equation and make it easier to solve. However, for this particular quintic equation, factoring is not straightforward. The rational root theorem can be used to test for possible rational roots, but it does not guarantee that we will find any. In this case, the possible rational roots would be the factors of 15 divided by the factors of 3, which are ±1, ±3, ±5, ±15, ±1/3, and ±5/3. Testing these values might reveal some rational roots, but it is also possible that all roots are irrational or complex. Due to the complexity of solving this quintic equation analytically, numerical or graphical methods are typically employed to find the approximate solutions. These methods provide a practical way to find the roots, even when an exact algebraic solution is not feasible. Understanding the limitations of analytical methods and the applicability of numerical and graphical methods is crucial in solving higher-degree polynomial equations.

Conclusion

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

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

This equation is obtained by setting the two expressions for ${ y }$ in the system equal to each other. Solving this equation will give us the ${ x }$-values where the graphs of the two equations intersect, which are the solutions to the system. The process of identifying the correct equation and understanding how it relates to the system is a fundamental concept in algebra. It demonstrates the power of systems of equations in solving more complex problems. By transforming a system into a single equation, we can often simplify the problem and apply various algebraic techniques to find the solution. The resulting quintic equation, 3x⁵ - 4x⁴ - 11x³ + 2x² - 10x + 15 = 0, is challenging to solve analytically, highlighting the limitations of purely algebraic methods for higher-degree polynomials. Numerical and graphical methods provide practical alternatives for approximating the solutions in such cases. These methods are essential tools in mathematical problem-solving, allowing us to find solutions even when exact algebraic solutions are not feasible. This article has walked through the steps of transforming a system of equations into a single equation, identifying the correct equation from a set of options, and discussing the methods for solving the resulting polynomial equation. This approach not only answers the specific question but also provides a broader understanding of the principles and techniques involved in solving mathematical problems. By mastering these concepts, you can confidently tackle similar challenges and enhance your problem-solving skills in mathematics. The ability to manipulate equations, recognize patterns, and apply appropriate methods is crucial for success in mathematics and related fields. This article serves as a stepping stone towards developing these skills and fostering a deeper appreciation for the elegance and power of mathematical reasoning. The journey through this problem underscores the interconnectedness of various mathematical concepts and the importance of a holistic approach to problem-solving. By understanding the underlying principles and techniques, you can unlock the potential to solve a wide range of mathematical problems and contribute to the advancement of knowledge in this fascinating field.