Solving Systems Of Equations A Step-by-Step Guide To Solving Y=2x^2 And Y=-3x-1

In mathematics, solving a system of equations involves finding the values that satisfy all equations simultaneously. This article delves into the step-by-step process of solving a system of equations, particularly focusing on a non-linear system. We will address the specific system:

$y = 2x^2$ $y = -3x - 1$

This system combines a quadratic equation ($y = 2x^2$) and a linear equation ($y = -3x - 1$), presenting an interesting challenge. Let's explore the methods and techniques to find the solution(s) to this system.

Understanding the Problem

Before diving into the solution, it’s crucial to understand the nature of the equations. The first equation, y = 2x^2, represents a parabola, a U-shaped curve that opens upwards. The second equation, y = -3x - 1, is a linear equation, representing a straight line with a slope of -3 and a y-intercept of -1. Solving this system means finding the points where the parabola and the line intersect. These points of intersection represent the (x, y) values that satisfy both equations.

Graphically, the solutions are the coordinates of the points where the parabola and the line meet. There can be zero, one, or two intersection points in such a system. Algebraically, we find these points by using methods like substitution or elimination. In this case, substitution is the most straightforward approach due to the nature of the given equations. By setting the two expressions for y equal to each other, we can create a single equation in terms of x. This equation can then be solved using algebraic techniques such as factoring, completing the square, or the quadratic formula. Understanding the graphical representation alongside the algebraic method provides a more intuitive grasp of the solution process, enhancing the problem-solving experience and ensuring accuracy in the results.

Step-by-Step Solution

Step 1: Substitution Method

The given system of equations is:

$y = 2x^2$ (Equation 1) $y = -3x - 1$ (Equation 2)

Since both equations are expressed in terms of y, we can use the substitution method. This involves setting the expressions for y from both equations equal to each other. By doing this, we create a new equation that only involves the variable x, simplifying the process of finding the x values that satisfy both original equations.

Equating Equation 1 and Equation 2, we get:

$2x^2 = -3x - 1$

This equation is a quadratic equation, which can be solved using several methods. The substitution method is particularly useful here as it transforms a system of two equations into a single, solvable equation. The next step is to rearrange the equation into a standard quadratic form, which is essential for applying methods like factoring or the quadratic formula. Understanding the substitution method and its application in solving systems of equations is crucial for tackling more complex problems in algebra and beyond.

Step 2: Rearrange into Quadratic Form

To solve the equation $2x^2 = -3x - 1$, we first need to rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. This form is essential because it allows us to easily apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Moving all terms to one side of the equation sets the stage for these solution methods.

Adding $3x$ and $1$ to both sides of the equation, we get:

$2x^2 + 3x + 1 = 0$

This is now a quadratic equation in standard form, where $a = 2$, $b = 3$, and $c = 1$. Identifying the coefficients a, b, and c is a critical step, especially if we plan to use the quadratic formula. Recognizing the equation in this form also helps in deciding the most appropriate method for solving it. For instance, if the equation can be easily factored, that would be the quickest route to the solution. If not, the quadratic formula provides a reliable alternative. The ability to manipulate and rearrange equations into standard forms is a fundamental skill in algebra, providing a foundation for solving a wide range of mathematical problems.

Step 3: Solve the Quadratic Equation

Now that we have the quadratic equation in the standard form $2x^2 + 3x + 1 = 0$, we can proceed to solve it. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a straightforward approach.

Factoring the Quadratic Equation

Factoring involves breaking down the quadratic expression into two binomial factors. We look for two numbers that multiply to give the product of $a$ and $c$ (in this case, $2 imes 1 = 2$) and add up to $b$ (which is 3). The numbers 1 and 2 satisfy these conditions since $1 imes 2 = 2$ and $1 + 2 = 3$. We can rewrite the middle term using these numbers:

$2x^2 + 2x + x + 1 = 0$

Next, we factor by grouping:

$2x(x + 1) + 1(x + 1) = 0$

$(2x + 1)(x + 1) = 0$

Setting each factor equal to zero gives us the possible solutions for x:

$2x + 1 = 0$ or $x + 1 = 0$

Solving these linear equations, we find:

x = -rac{1}{2} or $x = -1$

These are the x-coordinates of the points where the parabola and the line intersect. To find the corresponding y-coordinates, we substitute these x values back into either of the original equations. Factoring is an efficient method when the quadratic equation is factorable, as it provides a direct route to the solutions. However, not all quadratic equations can be easily factored, making it essential to know alternative methods like the quadratic formula.

Step 4: Find the Corresponding y Values

We have found the x values where the line and parabola intersect: x = -rac{1}{2} and $x = -1$. To complete the solution, we need to find the corresponding y values for each x value. We can do this by substituting each x value back into either of the original equations. It is often simpler to use the linear equation ($y = -3x - 1$) for this purpose, as it involves less computation. However, using the quadratic equation ($y = 2x^2$) will yield the same results.

For x = -rac{1}{2}:

Substitute x = -rac{1}{2} into $y = -3x - 1$:

y = -3(-rac{1}{2}) - 1 y = rac{3}{2} - 1 y = rac{3}{2} - rac{2}{2} y = rac{1}{2}

So, one solution is (-rac{1}{2}, rac{1}{2}).

For $x = -1$:

Substitute $x = -1$ into $y = -3x - 1$:

$y = -3(-1) - 1$ $y = 3 - 1$ $y = 2$

Thus, another solution is $(-1, 2)$.

Finding the y values corresponding to the x values is a crucial step in solving systems of equations, as it provides the complete coordinates of the points of intersection. By substituting the x values back into the original equations, we ensure that the solutions satisfy both equations simultaneously. This step highlights the importance of verifying the solutions to ensure accuracy and completeness in the problem-solving process.

Step 5: State the Solutions

After finding both the x and y values, the final step is to clearly state the solutions to the system of equations. We have determined that the line and the parabola intersect at two points. The solutions are the coordinates of these points, which represent the pairs of x and y values that satisfy both equations simultaneously.

The solutions to the system of equations are:

(-rac{1}{2}, rac{1}{2}) and $(-1, 2)$

These points represent the intersection points of the parabola $y = 2x^2$ and the line $y = -3x - 1$. Graphically, these are the points where the two curves meet on the coordinate plane. Algebraically, substituting these x and y values into both equations will confirm that they hold true.

Stating the solutions clearly is essential for presenting the final answer in a comprehensive manner. It provides a concise summary of the results obtained through the step-by-step solution process. Furthermore, understanding how to present solutions effectively is an important skill in mathematics, ensuring clarity and accuracy in communication. Verifying the solutions by substituting them back into the original equations is a good practice to ensure the correctness of the answers and to reinforce the understanding of the solution process.

Conclusion

In conclusion, solving the system of equations

$y = 2x^2$ $y = -3x - 1$

involved using the substitution method to equate the expressions for y, rearranging the resulting equation into a standard quadratic form, solving the quadratic equation by factoring, and then finding the corresponding y values for each x value. This process led us to the solutions (-rac{1}{2}, rac{1}{2}) and $(-1, 2)$, which represent the points of intersection between the parabola and the line.

This exercise illustrates the importance of understanding various algebraic techniques, such as substitution, factoring, and solving quadratic equations, in finding solutions to systems of equations. The combination of a quadratic and a linear equation provides a practical example of how these methods can be applied in a non-linear context. The ability to solve such systems is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, and economics.

By following a structured approach and understanding the underlying principles, students can confidently tackle similar problems. The step-by-step solution presented here serves as a guide for solving systems of equations, highlighting the logical progression from problem setup to final solution. Mastery of these techniques not only enhances problem-solving skills but also fosters a deeper appreciation for the interconnectedness of mathematical concepts.