Solutions For System Of Equations -3x^2+4y=21 And 4x^2+y=10

Let's dive into the realm of solving systems of equations, a fundamental concept in mathematics with wide-ranging applications in various fields. In this comprehensive guide, we will tackle the specific system presented and, more broadly, explore the techniques and strategies for solving such problems. Our focus will be on understanding the underlying principles and applying them to find accurate solutions.

The Given System of Equations

We are presented with the following system of equations:

$ \begin{array}{l} -3 x^2+4 y=21 \ 4 x^2+y=10 \end{array} $

This system involves two equations with two variables, x and y. The first equation is a quadratic equation due to the presence of the $x^2$ term, while the second equation is also a quadratic equation due to the presence of the $x^2$ term but y only appears to the power of 1. Our objective is to find the values of x and y that satisfy both equations simultaneously. This means that when we substitute the values of x and y into both equations, the equations hold true.

Strategies for Solving Systems of Equations

There are several methods for solving systems of equations, each with its own advantages and applicability depending on the nature of the equations. Some common methods include:

1. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which can then be solved. The value obtained is then substituted back into one of the original equations to find the value of the other variable.

2. Elimination Method: This method involves manipulating the equations to eliminate one of the variables. This is typically done by multiplying one or both equations by constants so that the coefficients of one variable are opposites. Then, the equations are added together, eliminating that variable. The resulting equation with one variable can be solved, and the value is substituted back into one of the original equations to find the value of the other variable.

3. Graphical Method: This method involves graphing both equations on the same coordinate plane. The solutions to the system are the points where the graphs intersect. This method is particularly useful for visualizing the solutions, but it may not provide precise answers, especially for non-linear systems.

4. Matrix Methods: For systems with more than two variables, matrix methods such as Gaussian elimination and matrix inversion can be employed. These methods provide a systematic approach to solving linear systems.

Solving the System Using Elimination Method

For the given system, the elimination method seems like a straightforward approach. We can multiply the second equation by -4 to eliminate y:

Original System:

$ \begin{array}{l} -3 x^2+4 y=21 \ 4 x^2+y=10 \end{array} $

Multiply the second equation by -4:

$ \begin{array}{l} -3 x^2+4 y=21 \ -16 x^2-4y=-40 \end{array} $

Now, add the two equations together:

$(-3x^2 + 4y) + (-16x^2 - 4y) = 21 + (-40)$

This simplifies to:

$-19x^2 = -19$

Divide both sides by -19:

$x^2 = 1$

Take the square root of both sides:

$x = \pm 1$

So, we have two possible values for x: 1 and -1. Now, we need to find the corresponding values for y.

Finding the Values of y

We can substitute the values of x back into either of the original equations to solve for y. Let's use the second equation, as it seems simpler:

$4x^2 + y = 10$

Case 1: x = 1

Substitute x = 1 into the equation:

$4(1)^2 + y = 10$

$4 + y = 10$

$y = 10 - 4$

$y = 6$

So, one solution is (1, 6).

Case 2: x = -1

Substitute x = -1 into the equation:

$4(-1)^2 + y = 10$

$4 + y = 10$

$y = 10 - 4$

$y = 6$

So, another solution is (-1, 6).

Verifying the Solutions

It's crucial to verify our solutions by substituting them back into both original equations to ensure they hold true.

Solution 1: (1, 6)

Equation 1: $-3x^2 + 4y = 21$

$-3(1)^2 + 4(6) = -3 + 24 = 21$ (True)

Equation 2: $4x^2 + y = 10$

$4(1)^2 + 6 = 4 + 6 = 10$ (True)

Solution 2: (-1, 6)

Equation 1: $-3x^2 + 4y = 21$

$-3(-1)^2 + 4(6) = -3 + 24 = 21$ (True)

Equation 2: $4x^2 + y = 10$

$4(-1)^2 + 6 = 4 + 6 = 10$ (True)

Both solutions satisfy both equations, so they are valid.

Analyzing the Options

Now, let's look at the given options:

A. (1,6) and (0,10)

B. (-1,6) and (1,6)

C. (1,6) and (3,12)

D. (-1,6) and (3,12)

Our solutions are (1, 6) and (-1, 6). Comparing these with the options, we find that option B, (-1, 6) and (1, 6), matches our solutions.

Conclusion

Therefore, the solutions for the given system of equations are (-1, 6) and (1, 6). This comprehensive exploration has demonstrated the process of solving a system of quadratic equations using the elimination method, verifying the solutions, and identifying the correct answer from the given options. Understanding these techniques is essential for tackling more complex mathematical problems and real-world applications.

In summary, when dealing with systems of equations, it's vital to:

• Choose the appropriate method based on the equations' nature (substitution, elimination, graphical, or matrix methods).
• Carefully perform algebraic manipulations to isolate variables.
• Verify the solutions to ensure accuracy.
• Interpret the solutions in the context of the problem.

By mastering these steps, you'll be well-equipped to solve a wide array of mathematical challenges involving systems of equations.