Solve The Following System Of Equations:$\[ \begin{array}{l} y = X^2 + 10x + 35 \\ 4x + Y = 2 \end{array} \\]A. \[$(-11, 46)\$\] And \[$(3, 14)\$\] B. \[$(-11, 46)\$\] And \[$(-3, 14)\$\] C. \[$(-11,

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two equations, one quadratic and one linear, to find the values of the variables x and y.

The System of Equations

The given system of equations is:

$$\begin{array}{l} y = x^2 + 10x + 35 \\ 4x + y = 2 \end{array}$$

Our goal is to find the values of x and y that satisfy both equations.

Substitution Method

One way to solve this system of equations is by using the substitution method. We can substitute the expression for y from the first equation into the second equation.

$$4x + (x^2 + 10x + 35) = 2$$

Simplifying the equation, we get:

$$x^2 + 14x + 33 = 0$$

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 = 14, and c = 33. Plugging these values into the formula, we get:

$$x = \frac{-14 \pm \sqrt{14^2 - 4(1)(33)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-14 \pm \sqrt{196 - 132}}{2}$$

$$x = \frac{-14 \pm \sqrt{64}}{2}$$

$$x = \frac{-14 \pm 8}{2}$$

This gives us two possible values for x:

$$x = \frac{-14 + 8}{2} = -3$$

$$x = \frac{-14 - 8}{2} = -11$$

Finding the Values of y

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

For x = -3, we substitute this value into the first equation:

$$y = (-3)^2 + 10(-3) + 35$$

Simplifying the expression, we get:

$$y = 9 - 30 + 35$$

$$y = 14$$

For x = -11, we substitute this value into the first equation:

$$y = (-11)^2 + 10(-11) + 35$$

Simplifying the expression, we get:

$$y = 121 - 110 + 35$$

$$y = 46$$

Conclusion

Therefore, the solutions to the system of equations are:

$$(-11, 46)$$

$$(-3, 14)$$

These are the values of x and y that satisfy both equations.

Discussion

The system of equations can be solved using the substitution method, where we substitute the expression for y from the first equation into the second equation. This results in a quadratic equation, which can be solved using the quadratic formula. The solutions to the system of equations are the values of x and y that satisfy both equations.

Final Answer

The final answer is:

$$(-11, 46)$$

(-3, 14)$<br/> **Solving a System of Equations: A Q&A Guide** ===================================================== **Introduction** --------------- In our previous article, we solved a system of equations using the substitution method. In this article, we will answer some frequently asked questions about solving systems of equations. **Q: What is a system of equations?** ----------------------------------- A system of equations is a set of two or more equations that involve the same variables. In other words, it is a collection of equations that must be solved simultaneously to find the values of the variables. **Q: How do I know which method to use to solve a system of equations?** ------------------------------------------------------------------- There are several methods to solve a system of equations, including the substitution method, the elimination method, and the graphing method. The choice of method depends on the type of equations and the variables involved. **Q: What is the substitution method?** -------------------------------------- The substitution method involves substituting the expression for one variable from one equation into the other equation. This results in a new equation that can be solved for the remaining variable. **Q: What is the elimination method?** -------------------------------------- The elimination method involves adding or subtracting the equations to eliminate one of the variables. This results in a new equation that can be solved for the remaining variable. **Q: What is the graphing method?** ----------------------------------- The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. This point represents the solution to the system of equations. **Q: How do I know if a system of equations has a solution?** ------------------------------------------------------ A system of equations has a solution if the equations are consistent and the variables are related in a way that allows for a unique solution. If the equations are inconsistent or the variables are unrelated, there may be no solution or an infinite number of solutions. **Q: What is a consistent system of equations?** --------------------------------------------- A consistent system of equations is a system that has a solution. In other words, the equations are related in a way that allows for a unique solution. **Q: What is an inconsistent system of equations?** ---------------------------------------------- An inconsistent system of equations is a system that has no solution. In other words, the equations are unrelated and cannot be solved simultaneously. **Q: What is an infinite system of equations?** ---------------------------------------------- An infinite system of equations is a system that has an infinite number of solutions. In other words, the equations are related in a way that allows for multiple solutions. **Q: How do I know if a system of equations has an infinite number of solutions?** -------------------------------------------------------------------------------- A system of equations has an infinite number of solutions if the equations are dependent and the variables are related in a way that allows for multiple solutions. **Q: What is a dependent system of equations?** ---------------------------------------------- A dependent system of equations is a system that has an infinite number of solutions. In other words, the equations are related in a way that allows for multiple solutions. **Conclusion** ---------- Solving a system of equations can be a challenging task, but with the right methods and techniques, it can be done. In this article, we have answered some frequently asked questions about solving systems of equations and provided a guide to help you get started. **Final Answer** -------------- The final answer is: * A system of equations is a set of two or more equations that involve the same variables. * The substitution method involves substituting the expression for one variable from one equation into the other equation. * The elimination method involves adding or subtracting the equations to eliminate one of the variables. * The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. * A consistent system of equations is a system that has a solution. * An inconsistent system of equations is a system that has no solution. * An infinite system of equations is a system that has an infinite number of solutions. * A dependent system of equations is a system that has an infinite number of solutions.