Solve The System Of Equations:${ \begin{array}{l} x^2 + X = Y - 5 \ y - 5 = 2x + 2 \end{array} }$A. { (2, -1)$}$B. { (2, 11), (-1, 5)$}$C. { (-2, 3), (1, 9)$}$D. No Solution

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Introduction

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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 focus on solving a specific system of equations involving quadratic and linear equations. We will use algebraic methods to find the solutions and discuss the possible outcomes.

The System of Equations

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The given system of equations is:

$$\begin{array}{l} x^2 + x = y - 5 \\ y - 5 = 2x + 2 \end{array}$$

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

Step 1: Simplify the Second Equation

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Let's start by simplifying the second equation:

$$y - 5 = 2x + 2$$

We can add 5 to both sides to isolate $y$:

$$y = 2x + 7$$

Step 2: Substitute the Expression for $y$ into the First Equation

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Now, we can substitute the expression for $y$ into the first equation:

$$x^2 + x = (2x + 7) - 5$$

Simplifying the equation, we get:

$$x^2 + x = 2x + 2$$

Step 3: Rearrange the Equation

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To make it easier to solve, let's rearrange the equation by moving all terms to one side:

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

Combining like terms, we get:

$$x^2 - x - 2 = 0$$

Step 4: Solve the Quadratic Equation

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Now, we have a quadratic equation in the form of $ax^2 + bx + c = 0$. We can use the quadratic formula to find the solutions:

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

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

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

Simplifying the expression, we get:

$$x = \frac{1 \pm \sqrt{1 + 8}}{2}$$

$$x = \frac{1 \pm \sqrt{9}}{2}$$

$$x = \frac{1 \pm 3}{2}$$

This gives us two possible values for $x$:

$$x = \frac{1 + 3}{2} = 2$$

$$x = \frac{1 - 3}{2} = -1$$

Step 5: Find the Corresponding Values of $y$

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Now that we have the values of $x$, we can find the corresponding values of $y$ by substituting $x$ into one of the original equations. Let's use the second equation:

$$y = 2x + 7$$

For $x = 2$, we get:

$$y = 2(2) + 7 = 11$$

For $x = -1$, we get:

$$y = 2(-1) + 7 = 5$$

Conclusion

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In conclusion, we have found two possible solutions to the system of equations:

$$(2, 11)$$

$$( -1, 5)$$

Therefore, the correct answer is:

B. $(2, 11), (-1, 5)$

Note that there are two possible solutions, which means that the system of equations has two distinct solutions.

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Introduction

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In our previous article, we solved a system of equations involving quadratic and linear equations. We used algebraic methods to find the solutions and discussed the possible outcomes. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving systems of equations.

Q: What is a system of equations?

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A: A system of equations is a set of two or more equations that involve variables and constants. The goal is to find the values of the variables that satisfy all the equations simultaneously.

Q: What are the different types of systems of equations?

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A: There are two main types of systems of equations:

• Linear systems: These involve linear equations, which are equations in which the highest power of the variable is 1.
• Nonlinear systems: These involve nonlinear equations, which are equations in which the highest power of the variable is greater than 1.

Q: How do I solve a system of equations?

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A: There are several methods to solve a system of equations, including:

• Substitution method: This involves substituting one equation into another to eliminate one of the variables.
• Elimination method: This involves adding or subtracting equations to eliminate one of the variables.
• Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.
• Algebraic method: This involves using algebraic techniques, such as factoring and quadratic formula, to solve the system.

Q: What is the quadratic formula?

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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?

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

Q: What are the possible outcomes of solving a system of equations?

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A: The possible outcomes of solving a system of equations are:

• One solution: This occurs when the system has a unique solution.
• No solution: This occurs when the system has no solution.
• Infinitely many solutions: This occurs when the system has infinitely many solutions.

Q: How do I determine the number of solutions?

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A: To determine the number of solutions, you need to examine the graph of the system or use algebraic techniques to find the solutions.

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

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

• Not checking the solutions: Make sure to check the solutions to ensure they satisfy both equations.
• Not using the correct method: Choose the correct method to solve the system, such as substitution or elimination.
• Not simplifying the equations: Simplify the equations to make it easier to solve the system.

Conclusion

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In conclusion, solving systems of equations requires a combination of algebraic techniques and problem-solving skills. By understanding the different types of systems, methods of solution, and possible outcomes, you can better approach and solve systems of equations. Remember to check your solutions, choose the correct method, and simplify the equations to ensure accurate results.