Solving Systems Of Equations -2x^2+y=-5 And Y=-3x^2+5

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In the realm of mathematics, solving systems of equations is a fundamental skill with applications that extend far beyond the classroom. At its core, a system of equations represents a set of two or more equations that share common variables. The solutions to such a system are the values for these variables that simultaneously satisfy all equations within the system. These solutions represent the points where the graphs of the equations intersect, providing a visual representation of the algebraic solution. Understanding how to solve systems of equations is crucial not only for academic success but also for tackling real-world problems in fields like engineering, economics, and computer science.

This article delves into the process of finding solutions for a specific system of equations, offering a detailed, step-by-step approach to ensure clarity and comprehension. Our main question revolves around finding the solutions to the following system:

-2x^2 + y = -5
y = -3x^2 + 5

We will explore different methods for solving this system, focusing on the substitution method due to its efficiency and directness in this case. By the end of this article, you will not only know the correct solutions but also understand the underlying principles and techniques involved in solving similar systems of equations. This knowledge will empower you to confidently tackle a wide range of mathematical challenges.

Before diving into the solution, it's essential to fully understand the given system of equations. The system we are addressing consists of two equations:

  1. -2x² + y = -5
  2. y = -3x² + 5

Both equations are non-linear, meaning the variables have exponents other than 1. Specifically, these are quadratic equations due to the presence of the x² term. Equation 1 expresses a relationship between x², and y, while Equation 2 directly defines y in terms of x². The goal is to find the pair(s) of (x, y) values that satisfy both equations simultaneously.

Graphical Interpretation

Each equation in the system can be visualized as a curve on a coordinate plane. In this case, both equations represent parabolas. The solutions to the system correspond to the points where these parabolas intersect. The number of solutions (intersection points) can vary, with possibilities ranging from no solutions (the parabolas do not intersect) to one solution (the parabolas touch at a single point) to two solutions (the parabolas intersect at two distinct points). Understanding this graphical interpretation provides a valuable visual aid in comprehending the nature of the solutions.

Analytical Approach

To find the solutions analytically, we need to use algebraic methods to manipulate the equations and isolate the variables. Several methods can be employed, including substitution, elimination, and graphical methods. The choice of method often depends on the specific structure of the equations. In this case, the substitution method is particularly well-suited due to the presence of an equation that directly expresses y in terms of x² (Equation 2).

By carefully analyzing the equations and understanding their graphical representation, we can develop a strategic approach to solving the system and finding the values of x and y that satisfy both equations.

The substitution method is a powerful technique for solving systems of equations, especially when one equation is already solved for one variable in terms of the other. In our case, Equation 2 (y = -3x² + 5) is perfectly suited for substitution since it expresses y directly as a function of x². This allows us to substitute the expression for y from Equation 2 into Equation 1, effectively eliminating y and creating a single equation in terms of x.

Step-by-Step Substitution

  1. Substitute the expression for y from Equation 2 into Equation 1: -2x² + (-3x² + 5) = -5

  2. Simplify the equation by combining like terms: -2x² - 3x² + 5 = -5 -5x² + 5 = -5

  3. Isolate the term with x²: -5x² = -10

  4. Divide both sides by -5: x² = 2

  5. Solve for x by taking the square root of both sides: x = ±√2

This step is crucial as it reveals that there are two possible values for x: √2 and -√2. For each of these x-values, we need to find the corresponding y-value.

Finding the Corresponding y-values

Now that we have the x-values, we can substitute them back into either Equation 1 or Equation 2 to find the corresponding y-values. Equation 2 (y = -3x² + 5) is generally easier to work with in this case.

  1. Substitute x = √2 into Equation 2: y = -3(√2)² + 5 y = -3(2) + 5 y = -6 + 5 y = -1

  2. Substitute x = -√2 into Equation 2: y = -3(-√2)² + 5 y = -3(2) + 5 y = -6 + 5 y = -1

Notice that both x = √2 and x = -√2 yield the same y-value, y = -1. This means we have two solutions for the system of equations.

Based on the substitution method, we have found two potential solutions for the system of equations:

  1. (√2, -1)
  2. (-√2, -1)

To ensure the accuracy of our solutions, it's crucial to verify them by substituting the x and y values back into both original equations. If both equations are satisfied by the solution, then it is indeed a valid solution.

Verification for (√2, -1)

  1. Equation 1: -2x² + y = -5 -2(√2)² + (-1) = -5 -2(2) - 1 = -5 -4 - 1 = -5 -5 = -5 (Equation 1 is satisfied)

  2. Equation 2: y = -3x² + 5 -1 = -3(√2)² + 5 -1 = -3(2) + 5 -1 = -6 + 5 -1 = -1 (Equation 2 is satisfied)

Since (√2, -1) satisfies both equations, it is a valid solution.

Verification for (-√2, -1)

  1. Equation 1: -2x² + y = -5 -2(-√2)² + (-1) = -5 -2(2) - 1 = -5 -4 - 1 = -5 -5 = -5 (Equation 1 is satisfied)

  2. Equation 2: y = -3x² + 5 -1 = -3(-√2)² + 5 -1 = -3(2) + 5 -1 = -6 + 5 -1 = -1 (Equation 2 is satisfied)

Similarly, (-√2, -1) satisfies both equations, confirming it as a valid solution.

Conclusion of Solutions

Therefore, the solutions to the system of equations are (√2, -1) and (-√2, -1). These are the points where the parabolas represented by the two equations intersect on the coordinate plane.

Now that we have found the solutions to the system of equations, we can compare them to the provided answer choices to identify the correct option. The answer choices were:

A. (√5, -10) and (-√5, -10) B. (0, 2) C. (1, -2) D. (√2, -1) and (-√2, -1)

By comparing our solutions (√2, -1) and (-√2, -1) to the answer choices, it's clear that option D matches our solutions exactly. Therefore, option D is the correct answer.

The other options can be easily ruled out by simply plugging their respective values into the equations and observing whether they satisfy the condition.

  • Option A: When x = √5 and y = -10, Equation 2 gives us -10 = -3(√5)² + 5, which simplifies to -10 = -15 + 5, or -10 = -10. But, for Equation 1, it results in -2(√5)² + (-10) = -5, which simplifies to -20 = -5, this isn't true. Thus, A isn't correct.
  • Option B: When x = 0 and y = 2, Equation 2 results in 2 = -3(0)² + 5 which simplifies to 2 = 5, an untrue statement. Therefore, B isn't correct.
  • Option C: When x = 1 and y = -2, Equation 2 gives us -2 = -3(1)² + 5, which simplifies to -2 = 2, also an untrue statement. Thus, C isn't correct.

In this article, we embarked on a journey to solve a system of non-linear equations, demonstrating the power and elegance of mathematical techniques. We successfully identified the solutions to the system:

-2x^2 + y = -5
y = -3x^2 + 5

through a meticulous step-by-step process. The key to our success was the substitution method, which allowed us to eliminate one variable and transform the system into a single equation that was readily solvable. We found the solutions to be (√2, -1) and (-√2, -1), corresponding to option D in the given answer choices. These solutions were rigorously verified by substituting them back into the original equations, ensuring their accuracy and validity.

Beyond simply finding the solutions, this exploration highlighted several important concepts:

  1. Understanding the nature of equations: Recognizing the equations as parabolas provided a valuable graphical perspective on the solutions as intersection points.
  2. Strategic method selection: Choosing the substitution method was crucial for simplifying the problem and efficiently finding the solutions.
  3. Importance of verification: Verifying the solutions is an essential step in any mathematical problem-solving process, ensuring the correctness of the results.

Solving systems of equations is a fundamental skill in mathematics, with applications spanning various fields. By mastering techniques like substitution, elimination, and graphical methods, you can confidently tackle a wide range of problems. This article serves as a testament to the power of mathematical reasoning and the satisfaction of arriving at accurate and verified solutions. Remember to always approach problems systematically, carefully analyze the given information, and leverage the appropriate tools and techniques to achieve success. Practice regularly, and you'll find yourself becoming more proficient and confident in your mathematical abilities.