What Are The Solutions To The System Of Equations?${ \begin{array}{l} y = X^2 + 5x + 6 \ y = 3x + 6 \end{array} }$A. { (0, 6)$}$ And { (-2, 0)$}$ B. { (0, -3)$}$ And { (6, 0)$}$ C. [$(-3,

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Introduction

Solving a system of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. A system of equations is a set of two or more equations that involve the same variables. In this article, we will focus on solving a system of two equations with two variables. We will use algebraic methods to find the solutions to the system of equations.

The System of Equations

The system of equations we will be solving is given by:

y=x2+5x+6y = x^2 + 5x + 6

y=3x+6y = 3x + 6

Step 1: Equating the Two Equations

To solve the system of equations, we need to equate the two equations. This means that we set the two equations equal to each other:

x2+5x+6=3x+6x^2 + 5x + 6 = 3x + 6

Step 2: Simplifying the Equation

Now, we simplify the equation by combining like terms:

x2+2x=0x^2 + 2x = 0

Step 3: Factoring the Equation

We can factor the equation by finding the greatest common factor (GCF) of the two terms:

x(x+2)=0x(x + 2) = 0

Step 4: Solving for x

To solve for x, we need to find the values of x that make the equation true. We can do this by setting each factor equal to zero:

x=0x = 0

x+2=0x + 2 = 0

Solving for x, we get:

x=−2x = -2

Step 5: Finding the Corresponding y-Values

Now that we have the values of x, we can find the corresponding y-values by substituting the values of x into one of the original equations. We will use the first equation:

y=x2+5x+6y = x^2 + 5x + 6

Substituting x = 0, we get:

y=02+5(0)+6y = 0^2 + 5(0) + 6

y=6y = 6

Substituting x = -2, we get:

y=(−2)2+5(−2)+6y = (-2)^2 + 5(-2) + 6

y=4−10+6y = 4 - 10 + 6

y=0y = 0

Step 6: Writing the Solutions

The solutions to the system of equations are the points (x, y) that satisfy both equations. In this case, the solutions are:

(0,6)(0, 6)

(−2,0)(-2, 0)

Conclusion

In this article, we solved a system of two equations with two variables using algebraic methods. We equated the two equations, simplified the resulting equation, factored the equation, and solved for x. We then found the corresponding y-values by substituting the values of x into one of the original equations. The solutions to the system of equations are the points (x, y) that satisfy both equations.

Discussion

The solutions to the system of equations are the points (0, 6) and (-2, 0). These points represent the intersection of the two curves. The first point (0, 6) represents the point where the two curves intersect on the y-axis, while the second point (-2, 0) represents the point where the two curves intersect on the x-axis.

Final Answer

The final answer is:

(0,6)(0, 6)

(-2, 0)$<br/> # 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. We will cover topics such as the different methods of solving systems of equations, how to choose the correct method, and some common mistakes to avoid.

Q: What are the different methods of solving systems of equations?

A: There are several methods of solving systems of equations, including:

  • Substitution Method: This method involves substituting one equation into the other equation to solve for one variable.
  • Elimination Method: This method involves adding or subtracting the two equations to eliminate one variable.
  • Graphical Method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
  • Algebraic Method: This method involves using algebraic techniques such as factoring and solving quadratic equations to solve the system.

Q: How do I choose the correct method?

A: The choice of method depends on the type of system and the level of difficulty. For example:

  • Substitution Method: Use this method when one equation is easily solvable and can be substituted into the other equation.
  • Elimination Method: Use this method when the two equations have the same coefficient for one variable.
  • Graphical Method: Use this method when the system is simple and the graphs are easy to draw.
  • Algebraic Method: Use this method when the system is complex and requires advanced algebraic techniques.

Q: What are some common mistakes to avoid?

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

  • Not checking the solution: Make sure to check the solution by substituting the values back into both equations.
  • Not using the correct method: Choose the correct method for the type of system and level of difficulty.
  • Not simplifying the equations: Simplify the equations before solving to avoid unnecessary complications.
  • Not checking for extraneous solutions: Check for extraneous solutions by substituting the values back into both equations.

Q: How do I know if the solution is correct?

A: To check if the solution is correct, substitute the values back into both equations and make sure they are true. If the solution satisfies both equations, then it is correct.

Q: What if I get a quadratic equation?

A: If you get a quadratic equation, you can solve it using the quadratic formula or factoring. Make sure to check the solutions by substituting them back into both equations.

Q: Can I use a calculator to solve systems of equations?

A: Yes, you can use a calculator to solve systems of equations. However, make sure to check the solution by substituting the values back into both equations to ensure accuracy.

Q: How do I graph a system of equations?

A: To graph a system of equations, plot the two equations on a coordinate plane and find the point of intersection. Make sure to use a ruler or graph paper to draw the graphs accurately.

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

A: If you have a system with three or more equations, you can use the same methods as before, but you may need to use more advanced techniques such as matrices or determinants.

Conclusion

Solving systems of equations can be a bit challenging, but with practice and patience, you can master the different methods and techniques. Remember to choose the correct method for the type of system and level of difficulty, and make sure to check the solution by substituting the values back into both equations.

Final Answer

The final answer is:

  • Substitution Method: Substitute one equation into the other equation to solve for one variable.
  • Elimination Method: Add or subtract the two equations to eliminate one variable.
  • Graphical Method: Graph the two equations on a coordinate plane and find the point of intersection.
  • Algebraic Method: Use algebraic techniques such as factoring and solving quadratic equations to solve the system.

Note: The final answer is a summary of the different methods of solving systems of equations.