Select The Correct Answer.Solve The System Of Equations.$\[ \begin{array}{l} x^2 + Y^2 = 1 \\ y = -4x - 1 \end{array} \\]A. \[$\left(-\frac{8}{17}, \frac{15}{17}\right)\$\] And \[$(0, -1)\$\]B. \[$\left(-\frac{3}{5},

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving a system of equations that involves a quadratic equation and a linear equation. We will use algebraic methods to find the solutions to the system of equations.

The System of Equations

The given system of equations is:

$$\begin{array}{l} x^2 + y^2 = 1 \\ y = -4x - 1 \end{array}$$

We are asked to find the correct solution to this system of equations.

Step 1: Substitute the Linear Equation into the Quadratic Equation

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

$$x^2 + (-4x - 1)^2 = 1$$

Step 2: Expand and Simplify the Equation

Now, we will expand and simplify the equation.

$$x^2 + 16x^2 + 8x + 1 = 1$$

Combine like terms:

$$17x^2 + 8x = 0$$

Step 3: Factor the Quadratic Equation

We can factor the quadratic equation as follows:

$$x(17x + 8) = 0$$

Step 4: Solve for x

Now, we can solve for $x$ by setting each factor equal to zero.

$$x = 0 \quad \text{or} \quad 17x + 8 = 0$$

Solve for $x$ in the second equation:

$$17x = -8$$

$$x = -\frac{8}{17}$$

Step 5: Find the Corresponding Values of y

Now that we have found the values of $x$, we can find the corresponding values of $y$ by substituting the values of $x$ into the linear equation.

For $x = 0$:

$$y = -4(0) - 1$$

$$y = -1$$

For $x = -\frac{8}{17}$:

$$y = -4\left(-\frac{8}{17}\right) - 1$$

$$y = \frac{32}{17} - 1$$

$$y = \frac{32}{17} - \frac{17}{17}$$

$$y = \frac{15}{17}$$

Conclusion

In conclusion, the correct solution to the system of equations is:

$$\left(-\frac{8}{17}, \frac{15}{17}\right) \quad \text{and} \quad (0, -1)$$

Therefore, the correct answer is A.

Discussion

This problem requires the use of algebraic methods to solve a system of equations. The student must be able to substitute the linear equation into the quadratic equation, expand and simplify the equation, factor the quadratic equation, and solve for $x$. The student must also be able to find the corresponding values of $y$ by substituting the values of $x$ into the linear equation.

Tips and Tricks

• When solving a system of equations, it is often helpful to substitute the linear equation into the quadratic equation.
• When expanding and simplifying the equation, be sure to combine like terms.
• When factoring the quadratic equation, look for two binomials whose product is the quadratic expression.
• When solving for $x$, be sure to set each factor equal to zero.
• When finding the corresponding values of $y$, be sure to substitute the values of $x$ into the linear equation.

Practice Problems

1. Solve the system of equations:

$$\begin{array}{l} x^2 + y^2 = 4 \\ y = 2x - 1 \end{array}$$

2. Solve the system of equations:

$$\begin{array}{l} x^2 + y^2 = 9 \\ y = -3x + 2 \end{array}$$

3. Solve the system of equations:

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

Answer Key

1. $\left(1, 1\right)$ and $\left(-1, -1\right)$
2. $\left(1, 1\right)$ and $\left(-1, -1\right)$
3. $\left(2, 1\right)$ and $\left(-2, -1\right)$
   Solving Systems of Equations: A Q&A Guide =====================================================

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will provide a Q&A guide to help students understand and solve systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other. Each equation is a statement that two expressions are equal, and the system of equations is a collection of these statements.

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 substituting the expression for one variable from one equation 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 graph.

Q: How do I choose the correct method?

A: To choose the correct method, you need to analyze the equations and determine which method is most suitable. Consider the type of equations, the number of variables, and the complexity of the equations.

Q: What are some common mistakes to avoid?

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

• Not checking the solutions for consistency
• Not using the correct method for the type of equations
• Not simplifying the equations before solving
• Not checking for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to substitute the solutions back into the original equations and check if they are true. If the solutions do not satisfy the original equations, they are extraneous and should be discarded.

Q: What are some real-world applications of solving systems of equations?

A: Solving systems of equations has many real-world applications, including:

• Physics: Solving systems of equations is used to model the motion of objects and predict their behavior.
• Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Solving systems of equations is used to model economic systems and make predictions about economic behavior.

Conclusion

Solving systems of equations is a fundamental concept in mathematics, and it has many real-world applications. By understanding the different methods and techniques for solving systems of equations, you can become proficient in solving these types of problems. Remember to choose the correct method, check for extraneous solutions, and apply the concepts to real-world problems.

Practice Problems

1. Solve the system of equations:

$$\begin{array}{l} x^2 + y^2 = 4 \\ y = 2x - 1 \end{array}$$

2. Solve the system of equations:

$$\begin{array}{l} x^2 + y^2 = 9 \\ y = -3x + 2 \end{array}$$

3. Solve the system of equations:

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

Answer Key

1. $\left(1, 1\right)$ and $\left(-1, -1\right)$
2. $\left(1, 1\right)$ and $\left(-1, -1\right)$
3. $\left(2, 1\right)$ and $\left(-2, -1\right)$