How Many Real Solutions Does The System Have?$\[ \begin{array}{l} y = -3x - 1 \\ y = X^2 - 3x + 4 \end{array} \\]Enter Your Answer In The Box: \[$\square\$\]

Introduction

In this article, we will explore the concept of solving a system of equations and determine the number of real solutions it has. We will use a specific system of equations as an example to illustrate the process. The system consists of two equations: a linear equation and a quadratic equation.

Understanding the System of Equations

The system of equations is given by:

$$\begin{array}{l} y = -3x - 1 \\ y = x^2 - 3x + 4 \end{array}$$

To find the number of real solutions, we need to understand the nature of the two equations. The first equation is a linear equation, which represents a straight line on the coordinate plane. The second equation is a quadratic equation, which represents a parabola on the coordinate plane.

Solving the System of Equations

To solve the system of equations, we need to find the points of intersection between the two equations. We can do this by setting the two equations equal to each other and solving for x.

$$-3x - 1 = x^2 - 3x + 4$$

Rearranging the equation, we get:

$$x^2 - 6x + 5 = 0$$

This is a quadratic equation, and we can solve it using the quadratic formula:

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

In this case, a = 1, b = -6, and c = 5. Plugging these values into the formula, we get:

$$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(5)}}{2(1)}$$

Simplifying the equation, we get:

$$x = \frac{6 \pm \sqrt{36 - 20}}{2}$$

$$x = \frac{6 \pm \sqrt{16}}{2}$$

$$x = \frac{6 \pm 4}{2}$$

This gives us two possible values for x:

$$x = \frac{6 + 4}{2} = 5$$

$$x = \frac{6 - 4}{2} = 1$$

Finding the Corresponding y-Values

Now that we have the x-values, we can find the corresponding y-values by plugging them into one of the original equations. Let's use the first equation:

$$y = -3x - 1$$

Plugging in x = 5, we get:

$$y = -3(5) - 1 = -16$$

Plugging in x = 1, we get:

$$y = -3(1) - 1 = -4$$

Analyzing the Solutions

We have found two possible solutions: (5, -16) and (1, -4). To determine the number of real solutions, we need to analyze the nature of these solutions.

The Nature of the Solutions

The first solution, (5, -16), is a real solution because it satisfies both equations. The second solution, (1, -4), is also a real solution because it satisfies both equations.

Conclusion

In conclusion, the system of equations has two real solutions: (5, -16) and (1, -4). These solutions satisfy both equations and represent the points of intersection between the two equations.

Final Answer

$$

Introduction

In our previous article, we explored the concept of solving a system of equations and determined the number of real solutions it has. We used a specific system of equations as an example to illustrate the process. In this article, we will answer some frequently asked questions related to the system of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are related to each other. In our example, we had two equations:

$$\begin{array}{l} y = -3x - 1 \\ y = x^2 - 3x + 4 \end{array}$$

Q: How do I solve a system of equations?

To solve a system of equations, you need to find the points of intersection between the two equations. You can do this by setting the two equations equal to each other and solving for x. In our example, we set the two equations equal to each other and solved for x:

$$-3x - 1 = x^2 - 3x + 4$$

Q: What is the quadratic formula?

The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

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

In our example, we used the quadratic formula to solve the quadratic equation:

$$x^2 - 6x + 5 = 0$$

Q: How do I find the corresponding y-values?

Once you have found the x-values, you can find the corresponding y-values by plugging them into one of the original equations. In our example, we used the first equation to find the corresponding y-values:

$$y = -3x - 1$$

Q: What is the nature of the solutions?

The nature of the solutions depends on the type of equations you are dealing with. In our example, we had two real solutions: (5, -16) and (1, -4). These solutions satisfied both equations and represented the points of intersection between the two equations.

Q: How do I determine the number of real solutions?

To determine the number of real solutions, you need to analyze the nature of the solutions. In our example, we had two real solutions, which means that the system of equations had two real solutions.

Q: What is the final answer?

The final answer is: $\boxed{2}$

Conclusion

In conclusion, solving a system of equations involves finding the points of intersection between the two equations and determining the number of real solutions. We hope that this Q&A article has helped you understand the concept of solving a system of equations.

Additional Resources

If you want to learn more about solving systems of equations, we recommend checking out the following resources:

• Khan Academy: Solving Systems of Equations
• Mathway: Solving Systems of Equations
• Wolfram Alpha: Solving Systems of Equations

Final Thoughts

Solving systems of equations is an important concept in mathematics, and it has many real-world applications. We hope that this Q&A article has helped you understand the concept of solving a system of equations and has provided you with the tools and resources you need to solve systems of equations on your own.