What Is A System Of Equations For The Following Situation?A Group Of 12 People Went To See A Movie. The Cost To Go To The Movie Is \$10 For An Adult And \$6 For A Child. The Total Cost For The Group Was \$100.Choose The Correct System Of

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Introduction

A system of equations is a set of two or more equations that are used to solve for the values of variables. In this article, we will explore a real-world scenario where a system of equations can be used to solve for the number of adults and children who went to see a movie.

The Scenario

A group of 12 people went to see a movie. The cost to go to the movie is $10 for an adult and $6 for a child. The total cost for the group was $100. We need to find the number of adults and children who went to see the movie.

Setting Up the System of Equations

Let's use the variables A to represent the number of adults and C to represent the number of children. We know that the total number of people is 12, so we can write an equation:

A + C = 12

We also know that the total cost for the group was $100. Since the cost for an adult is $10 and the cost for a child is $6, we can write another equation:

10A + 6C = 100

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the elimination method. We can multiply the first equation by 6 to get:

6A + 6C = 72

Now we can subtract this equation from the second equation to get:

4A = 28

Dividing both sides by 4, we get:

A = 7

Now that we have the value of A, we can substitute it into the first equation to get:

7 + C = 12

Subtracting 7 from both sides, we get:

C = 5

Conclusion

In this article, we have seen how a system of equations can be used to solve for the number of adults and children who went to see a movie. We set up two equations based on the given information and used the elimination method to solve for the values of A and C. The final answer is A = 7 and C = 5.

Example Use Cases

  • Real-world applications: Systems of equations can be used to model real-world scenarios such as budgeting, finance, and resource allocation.
  • Science and engineering: Systems of equations can be used to model complex systems such as electrical circuits, mechanical systems, and population dynamics.
  • Computer science: Systems of equations can be used to solve problems in computer science such as linear programming, graph theory, and optimization.

Tips and Tricks

  • Use the elimination method: The elimination method is a powerful tool for solving systems of equations. It involves multiplying one or both equations by a constant to eliminate one of the variables.
  • Check your work: Always check your work by plugging the values back into the original equations to make sure they are true.
  • Use technology: There are many online tools and calculators that can help you solve systems of equations.

Common Mistakes

  • Not checking your work: Failing to check your work can lead to incorrect solutions.
  • Not using the elimination method: The elimination method is a powerful tool for solving systems of equations. Not using it can lead to more complex and time-consuming solutions.
  • Not using technology: Not using technology can lead to errors and incorrect solutions.

Conclusion

In conclusion, systems of equations are a powerful tool for solving problems in mathematics and real-world scenarios. By understanding how to set up and solve systems of equations, we can model complex systems and make informed decisions. Remember to use the elimination method, check your work, and use technology to ensure accurate solutions.

Final Answer

The final answer is A = 7 and C = 5.

Introduction

Systems of equations are a fundamental concept in mathematics and are used to solve problems in various fields such as science, engineering, and finance. In this article, we will answer some frequently asked questions about systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are used to solve for the values of variables. Each equation in the system is a statement that two or more variables are related in a specific way.

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

A: You have a system of equations if you have two or more equations that involve the same variables. For example, if you have two equations:

2x + 3y = 7

x - 2y = -3

Then you have a system of equations.

Q: What are the different types of systems of equations?

A: There are two main types of systems of equations:

  • Linear systems of equations: These are systems of equations where each equation is a linear equation. For example:

    2x + 3y = 7

    x - 2y = -3

  • Non-linear systems of equations: These are systems of equations where at least one equation is not a linear equation. For example:

    x^2 + 2y = 7

    x - 2y = -3

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including:

  • Substitution method: This involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination method: This involves adding or subtracting the equations to eliminate one of the variables.
  • Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the difference between a system of equations and a single equation?

A: A single equation is a statement that two or more variables are related in a specific way. A system of equations is a set of two or more equations that are used to solve for the values of variables.

Q: Can I have a system of equations with more than two variables?

A: Yes, you can have a system of equations with more than two variables. For example:

2x + 3y + z = 7

x - 2y + z = -3

x + y - z = 2

This is a system of three equations with three variables.

Q: Can I have a system of equations with more than two equations?

A: Yes, you can have a system of equations with more than two equations. For example:

2x + 3y = 7

x - 2y = -3

x + y = 2

This is a system of three equations with two variables.

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

A: A system of equations has a solution if the equations are consistent and the variables are related in a way that allows for a unique solution. If the equations are inconsistent or the variables are related in a way that allows for multiple solutions, then the system of equations does not have a solution.

Q: Can I have a system of equations with no solution?

A: Yes, you can have a system of equations with no solution. For example:

2x + 3y = 7

x - 2y = -3

x + y = 5

This system of equations has no solution because the equations are inconsistent.

Q: Can I have a system of equations with infinitely many solutions?

A: Yes, you can have a system of equations with infinitely many solutions. For example:

2x + 3y = 7

x + y = 2

This system of equations has infinitely many solutions because the equations are dependent.

Conclusion

In conclusion, systems of equations are a fundamental concept in mathematics and are used to solve problems in various fields. By understanding how to set up and solve systems of equations, we can model complex systems and make informed decisions. Remember to use the substitution method, elimination method, and graphical method to solve systems of equations, and to check your work to ensure accurate solutions.

Final Answer

The final answer is that systems of equations are a powerful tool for solving problems in mathematics and real-world scenarios. By understanding how to set up and solve systems of equations, we can model complex systems and make informed decisions.