David Wants To Find The Solutions Of This System Of Equations:${ \begin{array}{l} -4x - 7 = Y \ x^2 - 2x - 6 = Y \end{array} }$Which Statement Is True?A. There Are No Real Number Solutions.B. There Is One Unique Real Number Solution At

Introduction

In mathematics, a system of equations is a set of equations that are related to each other through a common variable or variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two equations with two variables, x and y.

The System of Equations

The system of equations given to us is:

{ \begin{array}{l} -4x - 7 = y \\ x^2 - 2x - 6 = y \end{array} \}

We are asked to find the solutions of this system of equations, which means we need to find the values of x and y that satisfy both equations.

Step 1: Write Down the Equations

The first step in solving a system of equations is to write down the equations. In this case, we have two equations:

1. $-4x - 7 = y$
2. $x^2 - 2x - 6 = y$

Step 2: Equate the Two Equations

Since both equations are equal to y, we can equate the two equations to each other:

$-4x - 7 = x^2 - 2x - 6$

Step 3: Simplify the Equation

Now, we can simplify the equation by combining like terms:

$x^2 + 2x - 13 = 0$

Step 4: Solve the Quadratic Equation

The equation $x^2 + 2x - 13 = 0$ is a quadratic equation, which can be solved using the quadratic formula:

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

In this case, a = 1, b = 2, and c = -13. Plugging these values into the quadratic formula, we get:

$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-13)}}{2(1)}$ $x = \frac{-2 \pm \sqrt{4 + 52}}{2}$ $x = \frac{-2 \pm \sqrt{56}}{2}$ $x = \frac{-2 \pm 2\sqrt{14}}{2}$ $x = -1 \pm \sqrt{14}$

Step 5: Find the Values of x

We have found two possible values of x:

$x = -1 + \sqrt{14}$ $x = -1 - \sqrt{14}$

Step 6: Find the Values of y

Now that we have found the values of x, we can substitute them into one of the original equations to find the values of y. Let's use the first equation:

$-4x - 7 = y$

Substituting $x = -1 + \sqrt{14}$, we get:

$-4(-1 + \sqrt{14}) - 7 = y$ $4 - 4\sqrt{14} - 7 = y$ $-3 - 4\sqrt{14} = y$

Substituting $x = -1 - \sqrt{14}$, we get:

$-4(-1 - \sqrt{14}) - 7 = y$ $4 + 4\sqrt{14} - 7 = y$ $-3 + 4\sqrt{14} = y$

Conclusion

We have found two possible solutions to the system of equations:

1. $x = -1 + \sqrt{14}$, $y = -3 - 4\sqrt{14}$
2. $x = -1 - \sqrt{14}$, $y = -3 + 4\sqrt{14}$

Therefore, the correct answer is:

A. There are no real number solutions.

Q: What is a system of equations?

A: A system of equations is a set of equations that are related to each other through a common variable or variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use various methods such as substitution, elimination, or graphing. The method you choose will depend on the type of equations and the number of variables involved.

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one equation 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 equations are linear and have the same coefficient for one variable.

Q: What is the graphing method?

A: The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and have a simple relationship.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions for solving systems of equations, such as the "solve" function.

Q: How do I know if a system of equations has a solution?

A: To determine if a system of equations has a solution, you can use the following methods:

• Check if the equations are consistent (i.e., they have the same solution).
• Check if the equations are inconsistent (i.e., they have no solution).
• Check if the equations are dependent (i.e., they have an infinite number of solutions).

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a set of equations where each equation is linear (i.e., it can be written in the form ax + by = c). A system of nonlinear equations is a set of equations where at least one equation is nonlinear (i.e., it cannot be written in the form ax + by = c).

Q: Can I solve a system of nonlinear equations using the same methods as a system of linear equations?

A: No, you cannot solve a system of nonlinear equations using the same methods as a system of linear equations. Nonlinear equations require different methods, such as numerical methods or graphical methods.

Q: What is the significance of solving systems of equations in real-world applications?

A: Solving systems of equations is essential in many real-world applications, such as:

• Physics and engineering: Solving systems of equations is used to model and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Solving systems of equations is used to model and analyze economic systems, such as supply and demand curves.
• Computer science: Solving systems of equations is used in computer graphics and game development to create realistic simulations.

Conclusion

Solving systems of equations is a fundamental concept in mathematics and has numerous applications in real-world scenarios. By understanding the different methods and techniques for solving systems of equations, you can tackle complex problems and make informed decisions in various fields.