The First Two Steps In Determining The Solution Set Of The System Of Equations, Y = X 2 − 6 X + 12 Y = X^2 - 6x + 12 Y = X 2 − 6 X + 12 And Y = 2 X − 4 Y = 2x - 4 Y = 2 X − 4 , Algebraically Are Shown In The Table Below:[\begin{tabular}{|c|c|}\hline\text{Step} & \text{Equation}

Introduction

Solving a system of equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore the first two steps in determining the solution set of a system of equations, using the given equations $y = x^2 - 6x + 12$ and $y = 2x - 4$ as an example.

Understanding the System of Equations

A system of equations is a set of two or more equations that contain the same variables. In this case, we have two equations:

1. $y = x^2 - 6x + 12$
2. $y = 2x - 4$

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

Step 1: Setting the Equations Equal to Each Other

The first step in solving a system of equations is to set the equations equal to each other. This is done by equating the two expressions for $y$:

$x^2 - 6x + 12 = 2x - 4$

By setting the equations equal to each other, we create a new equation that combines the two original equations.

Step 2: Simplifying the Equation

The next step is to simplify the equation by combining like terms:

$x^2 - 6x + 12 - 2x + 4 = 0$

This simplifies to:

$x^2 - 8x + 16 = 0$

We can further simplify the equation by factoring:

$(x - 4)^2 = 0$

This equation has a repeated root, which means that the solution set is a single point.

The Solution Set

To find the solution set, we need to find the values of $x$ and $y$ that satisfy the equation $(x - 4)^2 = 0$. Since the equation is a perfect square, we know that the only solution is $x = 4$.

Substituting $x = 4$ into one of the original equations, we get:

$y = 2(4) - 4$

$y = 4$

Therefore, the solution set is $(4, 4)$.

Conclusion

In this article, we have explored the first two steps in determining the solution set of a system of equations. By setting the equations equal to each other and simplifying the resulting equation, we can find the solution set. In this case, the solution set is a single point, $(4, 4)$. This is a fundamental concept in mathematics, and it has many practical applications in fields such as physics, engineering, and economics.

Future Steps

In the next article, we will explore the remaining steps in determining the solution set of a system of equations. We will discuss how to find the solution set when the equation has multiple solutions, and how to use graphical methods to visualize the solution set.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Table of Contents

1. Introduction
2. Understanding the System of Equations
3. Step 1: Setting the Equations Equal to Each Other
4. Step 2: Simplifying the Equation
5. The Solution Set
6. Conclusion
7. Future Steps
8. References
9. Table of Contents
   Frequently Asked Questions (FAQs) About Solving Systems of Equations ====================================================================

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it can be a bit challenging for some students. In this article, we will answer some frequently asked questions (FAQs) about solving systems of equations. Whether you are a student, teacher, or just someone who wants to learn more about this topic, this article is for you.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain the same variables. For example, the system of equations:

1. $y = x^2 - 6x + 12$
2. $y = 2x - 4$

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to follow these steps:

1. Set the equations equal to each other.
2. Simplify the resulting equation.
3. Solve for the variables.

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

A: A linear system of equations is a system of equations where each equation is a linear equation. For example:

1. $y = 2x - 4$
2. $y = 3x + 2$

A nonlinear system of equations is a system of equations where at least one equation is a nonlinear equation. For example:

1. $y = x^2 - 6x + 12$
2. $y = 2x - 4$

Q: How do I determine the number of solutions to a system of equations?

A: To determine the number of solutions to a system of equations, you need to look at the graph of the equations. If the graphs intersect at a single point, then the system has one solution. If the graphs intersect at multiple points, then the system has multiple solutions. If the graphs do not intersect, then the system has no solutions.

Q: What is the solution set of a system of equations?

A: The solution set of a system of equations is the set of all possible solutions to the system. For example, if the system of equations has one solution, then the solution set is a single point. If the system of equations has multiple solutions, then the solution set is a set of points.

Q: How do I graph a system of equations?

A: To graph a system of equations, you need to graph each equation separately and then find the intersection points of the graphs. You can use a graphing calculator or a computer program to graph the equations.

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

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

• Not setting the equations equal to each other.
• Not simplifying the resulting equation.
• Not solving for the variables.
• Not checking the solutions.

Conclusion

Solving systems of equations is a fundamental concept in mathematics, and it can be a bit challenging for some students. However, with practice and patience, you can become proficient in solving systems of equations. Remember to follow the steps outlined in this article, and don't be afraid to ask for help if you need it.

Additional Resources

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang
• [4] Khan Academy: Solving Systems of Equations
• [5] Mathway: Solving Systems of Equations

Table of Contents

1. Introduction
2. Q: What is a system of equations?
3. Q: How do I solve a system of equations?
4. Q: What is the difference between a linear and a nonlinear system of equations?
5. Q: How do I determine the number of solutions to a system of equations?
6. Q: What is the solution set of a system of equations?
7. Q: How do I graph a system of equations?
8. Q: What are some common mistakes to avoid when solving systems of equations?
9. Conclusion
10. Additional Resources
11. Table of Contents