Solve The Simultaneous Equations:$ \begin{array}{l} y = 9 - X \\ y = 2x^2 + 4x + 6 \end{array} $Write Each Set Of Answers On Separate Lines, E.g.:$ \begin{array}{l} x = 13, Y = 15 \\ x = -3, Y = -8 \end{array} $

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Introduction

Simultaneous equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to solve simultaneous equations, with a focus on the given problem: y=9−xy = 9 - x and y=2x2+4x+6y = 2x^2 + 4x + 6. We will break down the solution into manageable steps, making it easy to understand and apply.

Understanding the Problem

The problem consists of two equations:

  1. y=9−xy = 9 - x
  2. y=2x2+4x+6y = 2x^2 + 4x + 6

Our goal is to find the values of xx and yy that satisfy both equations simultaneously.

Step 1: Equating the Two Equations

To solve the simultaneous equations, we need to equate the two equations and eliminate one of the variables. We can do this by setting the two equations equal to each other:

9−x=2x2+4x+69 - x = 2x^2 + 4x + 6

Step 2: Rearranging the Equation

Next, we need to rearrange the equation to get all the terms on one side:

2x2+4x+6−9+x=02x^2 + 4x + 6 - 9 + x = 0

Simplifying the equation, we get:

2x2+5x−3=02x^2 + 5x - 3 = 0

Step 3: Solving the Quadratic Equation

Now, we need to solve the quadratic equation 2x2+5x−3=02x^2 + 5x - 3 = 0. We can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a=2a = 2, b=5b = 5, and c=−3c = -3. Plugging these values into the formula, we get:

x=−5±52−4(2)(−3)2(2)x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)}

Simplifying the expression, we get:

x=−5±25+244x = \frac{-5 \pm \sqrt{25 + 24}}{4}

x=−5±494x = \frac{-5 \pm \sqrt{49}}{4}

x=−5±74x = \frac{-5 \pm 7}{4}

Step 4: Finding the Values of xx

Now, we need to find the values of xx by solving the two possible equations:

x=−5+74x = \frac{-5 + 7}{4}

x=24x = \frac{2}{4}

x=12x = \frac{1}{2}

x=−5−74x = \frac{-5 - 7}{4}

x=−124x = \frac{-12}{4}

x=−3x = -3

Step 5: Finding the Values of yy

Now that we have the values of xx, we can substitute them into one of the original equations to find the values of yy. We will use the first equation: y=9−xy = 9 - x.

For x=12x = \frac{1}{2}:

y=9−12y = 9 - \frac{1}{2}

y=172y = \frac{17}{2}

For x=−3x = -3:

y=9−(−3)y = 9 - (-3)

y=12y = 12

Conclusion

In this article, we have solved the simultaneous equations y=9−xy = 9 - x and y=2x2+4x+6y = 2x^2 + 4x + 6. We have broken down the solution into manageable steps, making it easy to understand and apply. We have found two sets of values for xx and yy that satisfy both equations simultaneously:

x=12,y=172x = \frac{1}{2}, y = \frac{17}{2}

x=−3,y=12x = -3, y = 12

We hope this article has provided a clear and concise guide to solving simultaneous equations. With practice and patience, you will become proficient in solving these types of equations and be able to apply them to a wide range of problems.

Final Answer

The final answer is:

x=12,y=172x = \frac{1}{2}, y = \frac{17}{2}

x=−3,y=12x = -3, y = 12

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Introduction

In our previous article, we explored how to solve simultaneous equations, with a focus on the given problem: y=9−xy = 9 - x and y=2x2+4x+6y = 2x^2 + 4x + 6. We broke down the solution into manageable steps, making it easy to understand and apply. In this article, we will answer some of the most frequently asked questions about solving simultaneous equations.

Q&A

Q: What are simultaneous equations?

A: Simultaneous equations are two or more equations that involve the same variables and are equal to each other. In other words, they are equations that have the same solution.

Q: How do I know which method to use to solve simultaneous equations?

A: There are several methods to solve simultaneous equations, including substitution, elimination, and graphing. The method you choose will depend on the type of equations you are working with and the level of difficulty.

Q: What is the substitution method?

A: The substitution method involves substituting one equation into the other equation to eliminate one of the variables. This method is useful when one of the equations is linear and the other is quadratic.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the two equations to eliminate one of the variables. This method is useful when the coefficients of the variables are the same in both equations.

Q: What is the graphing method?

A: The graphing method involves graphing both equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and the solution is a single point.

Q: How do I know if I have found the correct solution?

A: To check if you have found the correct solution, substitute the values of xx and yy back into both original equations. If the equations are true, then you have found the correct solution.

Q: What if I have a system of three or more equations?

A: If you have a system of three or more equations, you can use the same methods as before, but you may need to use a combination of methods or use a computer algebra system to solve the system.

Q: Can I use a calculator to solve simultaneous equations?

A: Yes, you can use a calculator to solve simultaneous equations. Many calculators have built-in functions for solving systems of equations, including the substitution and elimination methods.

Q: What if I get stuck or make a mistake?

A: If you get stuck or make a mistake, don't worry! Take a step back and review the problem. Check your work and make sure you have followed the correct steps. If you are still having trouble, try a different method or ask for help.

Conclusion

Solving simultaneous equations can be a challenging task, but with practice and patience, you will become proficient in solving these types of equations. Remember to choose the right method for the problem, check your work, and don't be afraid to ask for help. With this Q&A guide, you will be well on your way to becoming a master of solving simultaneous equations.

Final Tips

  • Practice, practice, practice! The more you practice solving simultaneous equations, the more comfortable you will become with the different methods.
  • Use a calculator or computer algebra system to check your work and ensure that you have found the correct solution.
  • Don't be afraid to ask for help if you are stuck or make a mistake.
  • Review the problem and check your work carefully before moving on to the next step.

Final Answer

The final answer is:

  • The substitution method is useful when one of the equations is linear and the other is quadratic.
  • The elimination method is useful when the coefficients of the variables are the same in both equations.
  • The graphing method is useful when the equations are linear and the solution is a single point.
  • You can use a calculator to solve simultaneous equations.
  • Don't be afraid to ask for help if you are stuck or make a mistake.