Solving Movie Attendance With Math: A System Of Equations Approach

Introduction

In this article, we will dive into an interesting mathematical problem involving movie attendance. Imagine a scenario where a movie theater hosted 500 people, with adults paying $10 each and children paying $4 each. The total revenue collected was $4160. Our mission is to determine the number of adults and children who attended the movie. This is a classic problem that can be solved using a system of linear equations. Let's embark on this mathematical journey together and unravel the mystery of movie attendance!

Setting Up the Equations

To solve this problem, we need to translate the given information into mathematical equations. Let's use $x$ to represent the number of adults and $y$ to represent the number of children who attended the movie. We can form two equations based on the given information:

1. Total Attendance: The total number of people who attended the movie is 500. This can be represented by the equation:

   $x + y = 500$

2. Total Revenue: The total amount collected from ticket sales is $4160. Since adults paid $10 each and children paid $4 each, we can represent this information with the following equation:

   $10x + 4y = 4160$

Now we have a system of two linear equations with two variables:

$x + y = 500$

$10x + 4y = 4160$

Solving the System of Equations

There are several methods to solve a system of linear equations, such as substitution, elimination, and graphing. In this case, we will use the substitution method. Here's how it works:

Step 1: Solve one equation for one variable

Let's solve the first equation ($x + y = 500$) for $y$:

$y = 500 - x$

Step 2: Substitute the expression into the other equation

Now, substitute this expression for $y$ into the second equation ($10x + 4y = 4160$):

$10x + 4(500 - x) = 4160$

Step 3: Simplify and solve for $x$

Distribute the 4 and simplify the equation:

$10x + 2000 - 4x = 4160$

Combine like terms:

$6x + 2000 = 4160$

Subtract 2000 from both sides:

$6x = 2160$

Divide both sides by 6:

$x = 360$

So, there were 360 adults who attended the movie.

Step 4: Substitute the value of $x$ back into the equation for $y$

Now that we know $x = 360$, we can substitute this value back into the equation $y = 500 - x$ to find the number of children:

$y = 500 - 360$

$y = 140$

Therefore, there were 140 children who attended the movie.

Verification

To ensure our solution is correct, we can substitute the values of $x$ and $y$ back into the original equations:

Equation 1: $x + y = 500$

$360 + 140 = 500$

$500 = 500$ (This is true)

Equation 2: $10x + 4y = 4160$

$10(360) + 4(140) = 4160$

$3600 + 560 = 4160$

$4160 = 4160$ (This is also true)

Since both equations hold true, our solution is correct. We have successfully determined that 360 adults and 140 children attended the movie.

Alternative Methods

While we used the substitution method to solve this system of equations, there are other methods we could have used. Let's briefly discuss two alternative approaches:

1. Elimination Method

The elimination method involves manipulating the equations so that one of the variables cancels out when the equations are added or subtracted. To use this method in our problem, we could multiply the first equation by -4 to eliminate $y$:

$-4(x + y) = -4(500)$

$-4x - 4y = -2000$

Now, add this modified equation to the second equation:

$(-4x - 4y) + (10x + 4y) = -2000 + 4160$

$6x = 2160$

$x = 360$

As we found before, $x = 360$. We can then substitute this value back into either of the original equations to solve for $y$.

2. Graphing Method

The graphing method involves plotting the two equations on a coordinate plane. The point where the two lines intersect represents the solution to the system of equations. To use this method, we would first rewrite each equation in slope-intercept form ($y = mx + b$):

$x + y = 500$ becomes $y = -x + 500$

$10x + 4y = 4160$ becomes $y = -2.5x + 1040$

By graphing these two lines, we would find that they intersect at the point (360, 140), confirming our solution of 360 adults and 140 children.

Real-World Applications

This type of problem, involving systems of linear equations, has numerous real-world applications. Here are a few examples:

1. Business and Finance

Businesses often use systems of equations to analyze costs, revenue, and profit. For example, a company might use a system of equations to determine the break-even point for a new product, where the total revenue equals the total cost.

2. Chemistry

In chemistry, systems of equations are used to balance chemical equations. Balancing equations ensures that the number of atoms of each element is the same on both sides of the equation, following the law of conservation of mass.

3. Nutrition

Dieticians and nutritionists use systems of equations to plan balanced diets. They can use equations to determine the amounts of different food groups needed to meet specific nutritional requirements, such as calorie, protein, and vitamin intake.

4. Engineering

Engineers use systems of equations in various applications, such as designing circuits, analyzing structures, and modeling fluid flow. These equations help engineers to predict the behavior of systems and optimize their designs.

Conclusion

In this article, we successfully solved a problem involving movie attendance using a system of linear equations. We determined that 360 adults and 140 children attended the movie, based on the given information about total attendance and revenue. We also explored alternative methods for solving systems of equations and discussed real-world applications of this mathematical concept. Understanding how to set up and solve systems of equations is a valuable skill that can be applied in various fields. Guys, I hope this article has helped you grasp the concept and its practical applications. Keep exploring the fascinating world of mathematics!