Consider The Following System Of Equations:${ \begin{array}{l} 10 + Y = 5x + X^2 \ 5x + Y = 1 \end{array} }$The First Equation Is An Equation Of A $\square$.The Second Equation Is An Equation Of A $\square$.

Mar 8, 2025 by ADMIN 213 views

**Solving a System of Equations: A Mathematical Exploration**

Introduction

In mathematics, solving a system of equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore a system of two equations and analyze the geometric representation of these equations. We will also discuss the methods used to solve this system and provide a step-by-step solution.

The System of Equations

The given system of equations is:

{ \begin{array}{l} 10 + y = 5x + x^2 \\ 5x + y = 1 \end{array} \}

The first equation is an equation of a parabola, which is a quadratic equation in the form of $y = ax^2 + bx + c$ . In this case, the equation is $y = -x^2 + 5x + 10$ . The second equation is an equation of a line, which is a linear equation in the form of $y = mx + b$ . In this case, the equation is $y = -5x + 1$ .

Geometric Representation

To visualize the system of equations, we can plot the parabola and the line on a coordinate plane. The parabola is a U-shaped curve that opens downwards, while the line is a straight line with a negative slope.

import numpy as np
import matplotlib.pyplot as plt
def parabola(x):
return -x**2 + 5*x + 10

def line(x):
return -5*x + 1

x = np.linspace(-10, 10, 400)

y_parabola = parabola(x)
y_line = line(x)

plt.plot(x, y_parabola, label='Parabola')
plt.plot(x, y_line, label='Line')
plt.xlabel('x')
plt.ylabel('y')
plt.title('System of Equations')
plt.legend()
plt.grid(True)
plt.show()

Methods for Solving the System

There are several methods for solving a system of equations, including:

Substitution Method: This method involves substituting the expression for one variable from one equation into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves plotting the equations on a coordinate plane and finding the point of intersection.

Step-by-Step Solution

To solve the system of equations, we can use the substitution method. We will substitute the expression for $y$ from the second equation into the first equation.

# Substitute y from the second equation into the first equation
# 10 + (-5x + 1) = 5x + x^2
# Simplify the equation
# -5x + 11 = 5x + x^2

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$ , $b = 10$ , and $c = -11$ . Plugging these values into the formula, we get:

$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-11)}}{2(1)}$ $x = \frac{-10 \pm \sqrt{100 + 44}}{2}$ $x = \frac{-10 \pm \sqrt{144}}{2}$ $x = \frac{-10 \pm 12}{2}$

Simplifying the expression, we get two possible values for $x$ :

$x_1 = \frac{-10 + 12}{2} = 1$ $x_2 = \frac{-10 - 12}{2} = -11$

Finding the Values of $y$

Now that we have found the values of $x$ , we can substitute these values into one of the original equations to find the corresponding values of $y$ .

Substituting $x_1 = 1$ into the second equation, we get:

$5(1) + y = 1$ $5 + y = 1$ $y = -4$

Substituting $x_2 = -11$ into the second equation, we get:

$5(-11) + y = 1$ $-55 + y = 1$ $y = 56$

Conclusion

In this article, we have explored a system of two equations and analyzed the geometric representation of these equations. We have also discussed the methods used to solve this system and provided a step-by-step solution. The system of equations can be solved using the substitution method, and the quadratic formula can be used to solve the resulting quadratic equation. The values of $x$ and $y$ can be found by substituting the values of $x$ into one of the original equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. In this article, we have a system of two equations with two variables, $x$ and $y$ .

Q: What are the different methods for solving a system of equations?

A: There are several methods for solving a system of equations, including:

Substitution Method: This method involves substituting the expression for one variable from one equation into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves plotting the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is linear and the other equation is quadratic.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable. This method is useful when the coefficients of the variables in the two equations are the same.

Q: What is the graphical method?

A: The graphical method involves plotting the equations on a coordinate plane and finding the point of intersection. This method is useful when the system of equations is simple and the graphs are easy to draw.

Q: How do I know which method to use?

A: The choice of method depends on the type of equations and the complexity of the system. If the equations are simple and the graphs are easy to draw, the graphical method may be the best choice. If the equations are more complex, the substitution or elimination method may be more suitable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . The formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$ , $b$ , and $c$ in the quadratic equation. Then, you plug these values into the formula and simplify the expression to find the values of $x$ .

Q: What are the steps to solve a system of equations?

A: The steps to solve a system of equations are:

Identify the equations: Identify the two equations in the system.
Choose a method: Choose a method to solve the system, such as substitution, elimination, or graphical.
Solve the system: Use the chosen method to solve the system and find the values of the variables.
Check the solution: Check the solution by plugging the values of the variables back into the original equations.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

Not checking the solution: Failing to check the solution by plugging the values of the variables back into the original equations.
Not using the correct method: Using the wrong method to solve the system, such as using the substitution method when the elimination method is more suitable.
Not simplifying the expression: Failing to simplify the expression when using the quadratic formula.

Q: How do I know if I have found the correct solution?

A: To know if you have found the correct solution, you need to check the solution by plugging the values of the variables back into the original equations. If the solution satisfies both equations, then it is the correct solution.