Solutions To System Of Equations 10x^2 - Y = 48 And 2y = 16x^2 + 48

In the realm of mathematics, systems of equations often present themselves as intriguing puzzles, challenging us to find the values that simultaneously satisfy multiple equations. This article delves into the process of solving a specific system of equations, providing a step-by-step approach to arrive at the solution. Our main focus will be on the system:

$$\left\{\begin{aligned} 10 x^2-y & =48 \\ 2 y & =16 x^2+48 \end{aligned}\right.$$

We will explore the methods used to find the values of $x$ and $y$ that make both equations true. This exploration will not only reveal the solution set but also illuminate the underlying mathematical principles involved. So, let's embark on this mathematical journey and unravel the mysteries of this system of equations.

Rewriting the Equations

To effectively tackle the system of equations, a strategic first step involves manipulating the equations into a more manageable form. Our initial focus is on isolating variables and creating opportunities for substitution or elimination. We begin with the given system:

$$\left\{\begin{aligned} 10 x^2-y & =48 \\ 2 y & =16 x^2+48 \end{aligned}\right.$$

The second equation, $2y = 16x^2 + 48$, presents a clear path for simplification. By dividing both sides of the equation by 2, we can isolate $y$:

$$y = 8x^2 + 24$$

This rewritten equation expresses $y$ in terms of $x$, setting the stage for substitution. We now have a clearer understanding of the relationship between $x$ and $y$, which will be crucial in the subsequent steps of solving the system. This methodical approach of simplifying equations is a cornerstone of algebraic problem-solving, allowing us to transform complex expressions into more workable forms.

Substitution Method

Now, with the equation $y = 8x^2 + 24$ at our disposal, we can employ the substitution method to solve the system. The core idea behind substitution is to replace one variable in an equation with its equivalent expression in terms of the other variable. In our case, we'll substitute the expression for $y$ from the rewritten second equation into the first equation.

The first equation is $10x^2 - y = 48$. Substituting $y = 8x^2 + 24$ into this equation, we get:

$$10x^2 - (8x^2 + 24) = 48$$

This substitution effectively eliminates $y$ from the equation, leaving us with an equation solely in terms of $x$. This is a significant step forward, as we can now focus on solving for a single variable. The resulting equation is a quadratic equation, which we can solve using various techniques such as factoring, completing the square, or the quadratic formula. The power of substitution lies in its ability to reduce a system of equations into a single equation, making the solution process more streamlined.

Solving for x

Having successfully substituted $y$ in the first equation, we now have a quadratic equation in terms of $x$. Our immediate goal is to simplify and solve this equation to find the possible values of $x$. Let's revisit the equation:

$$10x^2 - (8x^2 + 24) = 48$$

First, we distribute the negative sign:

$$10x^2 - 8x^2 - 24 = 48$$

Next, combine like terms:

$$2x^2 - 24 = 48$$

Add 24 to both sides:

$$2x^2 = 72$$

Divide both sides by 2:

$$x^2 = 36$$

Finally, take the square root of both sides:

$$x = \pm \sqrt{36}$$

$$x = \pm 6$$

This yields two possible values for $x$: $x = 6$ and $x = -6$. These values are crucial, as they represent the $x$-coordinates of the solutions to the system of equations. With these $x$ values in hand, we can proceed to find the corresponding $y$ values. The algebraic manipulations we've performed here are fundamental to solving quadratic equations and are a testament to the power of mathematical transformations.

Finding the Values of y

With the values of $x$ determined, our next task is to find the corresponding values of $y$. We'll utilize the equation we derived earlier, $y = 8x^2 + 24$, which expresses $y$ in terms of $x$. This equation serves as a direct link between the $x$ and $y$ values, allowing us to easily compute the $y$-coordinates of the solutions.

For $x = 6$, we substitute this value into the equation:

$$y = 8(6)^2 + 24$$

$$y = 8(36) + 24$$

$$y = 288 + 24$$

$$y = 312$$

So, when $x = 6$, $y = 312$. This gives us one solution to the system: $(6, 312)$.

Now, for $x = -6$, we substitute this value into the equation:

$$y = 8(-6)^2 + 24$$

$$y = 8(36) + 24$$

$$y = 288 + 24$$

$$y = 312$$

Interestingly, when $x = -6$, we also get $y = 312$. This gives us another solution to the system: $(-6, 312)$.

Thus, we have found two solutions to the system of equations. This process of back-substitution is a common technique in solving systems of equations, ensuring that we find the complete set of solutions.

The Solutions

Having meticulously worked through the system of equations, we have arrived at the solutions. Our final answer consists of the pairs $(x, y)$ that satisfy both equations simultaneously. We found that when $x = 6$, $y = 312$, and when $x = -6$, $y = 312$. Therefore, the solutions to the system are:

$$(6, 312) \text{ and } (-6, 312)$$

These points represent the intersections of the two curves defined by the equations in the system. Graphically, they are the points where the parabola $10x^2 - y = 48$ and the parabola $2y = 16x^2 + 48$ intersect. This complete solution set provides a comprehensive understanding of the relationship between the two equations and their common solutions.

Conclusion

In this article, we embarked on a journey to solve a system of equations, demonstrating the power of algebraic manipulation and the substitution method. We successfully found the solutions $(6, 312)$ and $(-6, 312)$, which satisfy both equations in the system. The key takeaways from this exploration include the importance of simplifying equations, strategically using substitution, and meticulously solving for each variable. Systems of equations are a fundamental concept in mathematics, and mastering the techniques to solve them is essential for various applications in science, engineering, and economics. This step-by-step approach not only provides the solution to this specific problem but also equips readers with the tools to tackle similar challenges in the future.