Solving Equations Using A System Of Equations
#system-of-equations #mathematics #algebra #equation-solving
This article delves into the process of identifying the equation that can be solved using a given system of equations. We will explore the underlying principles of systems of equations and how they relate to single equations. Understanding this connection is crucial for solving various mathematical problems efficiently. This article aims to provide a comprehensive explanation suitable for students and anyone interested in enhancing their algebraic skills.
Understanding Systems of Equations
In mathematics, a system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values that, when substituted for the variables, make all the equations in the system true. Solving systems of equations is a fundamental concept in algebra and finds applications in various fields, including engineering, physics, and economics. The beauty of a system of equations lies in its ability to represent real-world scenarios where multiple conditions or constraints are present simultaneously. The equations within the system interact, and finding the solution means identifying the values that satisfy all conditions at once.
When we talk about solving a system of equations, we're essentially seeking the point(s) where the graphs of the equations intersect. This intersection represents the values of the variables that satisfy all equations in the system. Consider a simple system of two linear equations in two variables. Each equation represents a line, and the solution is the point where these lines cross each other. If the lines are parallel, there's no solution because they never intersect. If they coincide, there are infinitely many solutions, as every point on the line satisfies both equations. The same principle extends to more complex systems involving non-linear equations. For example, in the given problem, we have a system consisting of a cubic equation and a quartic equation. The solutions to this system are the x-values where the graphs of these two equations intersect. The process of solving such systems often involves algebraic manipulation to eliminate variables and simplify the problem. Techniques such as substitution, elimination, and matrix methods are commonly used to solve systems of equations. Understanding these methods and their applications is crucial for tackling more advanced mathematical problems.
In the given system:
\left\{
\begin{array}{l}
y=3 x^3-7 x^2+5 \\
y=7 x^4+2 x
\end{array}
\right.
We have two equations. The first equation, y = 3x³ - 7x² + 5, represents a cubic function, while the second equation, y = 7x⁴ + 2x, represents a quartic function. To solve this system, we need to find the values of x and y that satisfy both equations simultaneously. This means we're looking for the points where the graphs of these two functions intersect. In essence, a system of equations is a powerful tool for modeling and solving problems involving multiple related quantities. By expressing the relationships between these quantities as equations, we can use algebraic techniques to find the values that satisfy all conditions. The solutions to the system provide valuable insights into the problem being modeled and can be used to make predictions or decisions.
Identifying the Equivalent Equation
The core concept in solving a system of equations is to find the values of the variables that satisfy all equations concurrently. To achieve this, we can often manipulate the equations to eliminate one or more variables, thereby reducing the system to a single equation in one variable. This single equation can then be solved using standard algebraic techniques. Let's delve into how this applies to the given system.
Consider the given system of equations:
\left\{
\begin{array}{l}
y=3 x^3-7 x^2+5 \\
y=7 x^4+2 x
\end{array}
\right.
Both equations are expressed in terms of y. This provides a direct pathway to eliminate y and obtain a single equation in terms of x. Since both equations are equal to y, we can equate the right-hand sides of the equations. This is based on the transitive property of equality, which states that if a = b and b = c, then a = c. In our case, if y = 3x³ - 7x² + 5 and y = 7x⁴ + 2x, then we can say that 3x³ - 7x² + 5 = 7x⁴ + 2x. This single equation now represents the condition that both the cubic and quartic functions have the same y-value for a particular x-value. Solving this equation will give us the x-coordinates of the points where the graphs of the two functions intersect. The resulting equation, 3x³ - 7x² + 5 = 7x⁴ + 2x, is a polynomial equation in one variable, x. To solve it, we would typically rearrange the terms to set the equation equal to zero and then use numerical or algebraic methods to find the roots. These roots represent the x-values that satisfy both equations in the original system. Equating the expressions allows us to combine the information from both equations into a single equation that captures the relationship between x and y. This approach is a fundamental technique in solving systems of equations and is applicable to a wide range of problems. By reducing the system to a single equation, we can simplify the problem and make it more manageable to solve.
Therefore, by setting the two expressions for y equal to each other, we obtain the equation:
3x³ - 7x² + 5 = 7x⁴ + 2x
This equation represents the condition where the two functions have the same y-value for a given x-value. Solving this equation will yield the x-coordinates of the points of intersection between the two curves represented by the original equations. This approach of equating expressions is a powerful technique in solving systems of equations. It allows us to combine the information from multiple equations into a single equation that captures the relationships between the variables. By solving this single equation, we can find the values of the variables that satisfy all equations in the system.
Analyzing the Answer Choices
Now, let's analyze the given answer choices in the context of the equation we derived:
A. 3x³ - 7x² + 5 = 0
This equation only considers the first equation in the system, setting it equal to zero. It does not incorporate the information from the second equation, which is crucial for solving the system. This equation represents finding the roots of the cubic function y = 3x³ - 7x² + 5, but it doesn't account for the condition that the solution must also satisfy the quartic equation y = 7x⁴ + 2x. Therefore, this option is not the correct equation that can be solved using the system of equations. Setting a single equation to zero is a common technique for finding its roots, but in the context of a system of equations, we need an equation that represents the intersection of the functions defined by the equations in the system. This requires considering both equations simultaneously.
B. 3x³ - 7x² + 5 = 7x⁴ + 2x
This equation is precisely what we derived by equating the two expressions for y. It represents the condition where the y-values of both equations are equal, which is the key to solving the system. This equation captures the essence of the problem by combining the information from both equations into a single equation. The solutions to this equation will give us the x-coordinates of the points where the graphs of the two functions intersect. This equation is the correct choice as it represents the equality of the two expressions for y, which is the fundamental step in solving the system of equations.
C. 7x⁴ + 2x = 0
Similar to option A, this equation only considers the second equation in the system, setting it equal to zero. It does not incorporate the information from the first equation. This equation represents finding the roots of the quartic function y = 7x⁴ + 2x, but it doesn't ensure that the solution also satisfies the cubic equation y = 3x³ - 7x² + 5. Therefore, this option is not the correct equation for solving the system. While finding the roots of individual equations can be a useful step in some problem-solving scenarios, it doesn't directly address the problem of finding the common solutions to a system of equations.
D. 7x⁴ + 3x³ - 7x² + 2x + 5 = 0
This equation is obtained by rearranging the terms in equation B to set it equal to zero: 7x⁴ - 3x³ + 7x² + 2x - 5 = 0. So, it represents the same condition as equation B, just in a rearranged form. This is achieved by moving all terms to one side of the equation, setting the other side to zero. This form is often preferred for solving polynomial equations, as it allows us to use techniques like factoring, the rational root theorem, or numerical methods to find the solutions. The solutions to this equation are the same as the solutions to equation B, as they both represent the same mathematical relationship. This equation is a valid form for solving the system of equations. To obtain the equation, you would subtract the cubic expression from both sides: (7x^4 + 2x) - (3x^3 - 7x^2 + 5) = 0 which simplifies to 7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0. The given option is incorrect because the signs are not correctly distributed during subtraction.
Conclusion
Based on our analysis, the correct equation that can be solved using the given system of equations is:
B. 3x³ - 7x² + 5 = 7x⁴ + 2x
This equation represents the condition where the two functions have the same y-value for a given x-value, which is the fundamental concept in solving a system of equations. Understanding how to manipulate and combine equations within a system is crucial for solving a wide range of mathematical problems. The process of equating expressions, as demonstrated in this article, is a powerful technique that can be applied to various scenarios involving systems of equations. By mastering these techniques, students and math enthusiasts can enhance their problem-solving skills and gain a deeper appreciation for the elegance and power of algebra. The ability to solve systems of equations is a valuable skill that extends beyond the classroom and finds applications in many real-world situations.
