The First Two Steps In Determining The Solution Set Of The System Of Equations, Y = − X 2 + 4 X + 12 Y=-x^2+4x+12 Y = − X 2 + 4 X + 12 And Y = − 3 X + 24 Y=-3x+24 Y = − 3 X + 24 , 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 focus on the first two steps in determining the solution set of a system of equations, using the given equations $y=-x^2+4x+12$ and $y=-3x+24$ as an example.

Step 1: Setting Up the Equations

The first step in solving a system of equations is to set up the equations in a way that allows us to find the solution set. In this case, we have two equations:

1. $y=-x^2+4x+12$
2. $y=-3x+24$

We can start by setting the two equations equal to each other, since both equations are equal to $y$. This gives us:

$$-x^2+4x+12=-3x+24$$

Step 2: Simplifying the Equation

The next step is to simplify the equation by combining like terms. We can start by moving all the terms to one side of the equation:

$$-x^2+4x+12+3x-24=0$$

This simplifies to:

$$-x^2+7x-12=0$$

We can further simplify the equation by factoring out a negative sign:

$$x^2-7x+12=0$$

The Quadratic Formula

The equation $x^2-7x+12=0$ is a quadratic equation, which can be solved using the quadratic formula:

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

In this case, $a=1$, $b=-7$, and $c=12$. Plugging these values into the formula, we get:

$$x=\frac{-(-7)\pm\sqrt{(-7)^2-4(1)(12)}}{2(1)}$$

This simplifies to:

$$x=\frac{7\pm\sqrt{49-48}}{2}$$

$$x=\frac{7\pm\sqrt{1}}{2}$$

$$x=\frac{7\pm1}{2}$$

Solving for x

We now have two possible solutions for $x$:

$$x=\frac{7+1}{2}=\frac{8}{2}=4$$

$$x=\frac{7-1}{2}=\frac{6}{2}=3$$

Conclusion

In this article, we have shown the first two steps in determining the solution set of a system of equations. We started by setting up the equations and simplifying the resulting equation. We then used the quadratic formula to solve for $x$. The two possible solutions for $x$ are $x=4$ and $x=3$. In the next article, we will continue with the solution set of the system of equations.

Future Steps

In the next article, we will:

• Solve for $y$ using the two possible values of $x$
• Determine the solution set of the system of equations
• Discuss the implications of the solution set

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart

Table of Contents

• Introduction
• Step 1: Setting Up the Equations
• Step 2: Simplifying the Equation
• The Quadratic Formula
• Solving for x
• Conclusion
• Future Steps
• References
• Table of Contents
  Frequently Asked Questions (FAQs) about Solving a System of Equations ====================================================================

Introduction

In our previous article, we discussed the first two steps in determining the solution set of a system of equations. We set up the equations, simplified the resulting equation, and used the quadratic formula to solve for $x$. In this article, we will answer some frequently asked questions (FAQs) about solving a system of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are equal to each other. In other words, it is a set of equations that have the same variable(s) and are related to each other.

Q: How do I know if I have a system of equations?

You have a system of equations if you have two or more equations that are equal to each other. For example:

1. $y=-x^2+4x+12$
2. $y=-3x+24$

These two equations are equal to each other, so they form a system of equations.

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

The solution set of a system of equations is the set of all possible values of the variables that satisfy the system of equations. In other words, it is the set of all possible solutions to the system of equations.

Q: How do I find the solution set of a system of equations?

To find the solution set of a system of equations, you need to follow these steps:

1. Set up the equations
2. Simplify the resulting equation
3. Use the quadratic formula to solve for $x$
4. Solve for $y$ using the two possible values of $x$
5. Determine the solution set of the system of equations

Q: What is the quadratic formula?

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

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

Q: How do I use the quadratic formula?

To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. For example, if you have the equation $x^2-7x+12=0$, you can plug in the values $a=1$, $b=-7$, and $c=12$ into the formula.

Q: What are the two possible solutions for $x$?

The two possible solutions for $x$ are given by:

$$x=\frac{7+1}{2}=\frac{8}{2}=4$$

$$x=\frac{7-1}{2}=\frac{6}{2}=3$$

Q: How do I determine the solution set of the system of equations?

To determine the solution set of the system of equations, you need to plug in the two possible values of $x$ into one of the original equations and solve for $y$. For example, if you plug in $x=4$ into the equation $y=-3x+24$, you get:

$$y=-3(4)+24=-12+24=12$$

So, the solution set of the system of equations is $(4,12)$.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about solving a system of equations. We have discussed the definition of a system of equations, the solution set of a system of equations, and the steps involved in finding the solution set. We have also discussed the quadratic formula and how to use it to solve quadratic equations.

Future Steps

In the next article, we will:

• Discuss the implications of the solution set of a system of equations
• Explore other methods for solving systems of equations
• Provide examples of real-world applications of systems of equations

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart

Table of Contents

• Introduction
• Q: What is a system of equations?
• Q: How do I know if I have a system of equations?
• Q: What is the solution set of a system of equations?
• Q: How do I find the solution set of a system of equations?
• Q: What is the quadratic formula?
• Q: How do I use the quadratic formula?
• Q: What are the two possible solutions for $x$?
• Q: How do I determine the solution set of the system of equations?
• Conclusion
• Future Steps
• References
• Table of Contents