Solving Systems Of Equations A Ski Rental Problem

Jul 12, 2025 by ADMIN 50 views

Understanding Systems of Equations in Ski and Snowboard Rentals

In the realm of mathematics, systems of equations provide a powerful tool for modeling and solving real-world problems. These systems involve two or more equations with multiple variables, allowing us to represent and analyze relationships between different quantities. One common application of systems of equations lies in the realm of business and finance, where they can be used to model costs, revenues, and profits. In this comprehensive article, we delve into the application of systems of equations to a ski rental store scenario, providing a step-by-step guide to formulating and solving these systems. By mastering this fundamental mathematical concept, readers can enhance their problem-solving skills and gain valuable insights into real-world applications.

Imagine a ski rental store that rents out both skis and snowboards. The store charges $44 for a pair of skis per day and $58 for a snowboard per day. On a particular day, the store made a total of $2,232 from ski and snowboard rentals. Additionally, the store rented out 9 more pairs of skis than snowboards. Our goal is to determine the number of skis and snowboards rented out on that day. This real-world scenario provides a perfect opportunity to apply the principles of systems of equations. By translating the given information into mathematical equations, we can systematically solve for the unknown quantities, gaining a clear understanding of the rental operations.

To effectively solve the ski rental problem, we need to translate the given information into a system of equations. This involves identifying the unknown variables and representing the relationships between them using mathematical expressions. Let's define our variables:

Let x represent the number of pairs of skis rented.
Let y represent the number of snowboards rented.

Now, let's translate the given information into equations:

Total Revenue: The store made $2,232 from ski and snowboard rentals. This can be expressed as an equation:
```
44x + 58y = 2232
```
This equation represents the total revenue generated from ski and snowboard rentals, where 44x is the revenue from skis and 58y is the revenue from snowboards.
Difference in Rentals: The store rented 9 more pairs of skis than snowboards. This can be expressed as an equation:
```
x = y + 9
```
This equation captures the relationship between the number of skis and snowboards rented, indicating that the number of skis rented (x) is 9 more than the number of snowboards rented (y).

Now we have a system of two equations:

44x + 58y = 2232
x = y + 9

This system of equations provides a mathematical representation of the ski rental scenario, allowing us to solve for the unknown variables x and y.

With the system of equations formulated, we can now employ various methods to solve for the unknown variables x and y. Two common methods for solving systems of equations are substitution and elimination. In this case, we will use the substitution method, as it is particularly well-suited for this problem.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved.

Solve for x in the second equation: The second equation, x = y + 9, is already solved for x.
Substitute into the first equation: Substitute the expression for x from the second equation into the first equation:
```
44(y + 9) + 58y = 2232
```
This substitution replaces x in the first equation with the expression y + 9, resulting in a single equation with only y as the variable.
Simplify and solve for y: Now, simplify and solve the equation for y:
```
44y + 396 + 58y = 2232
102y + 396 = 2232
102y = 1836
y = 18
```
This process involves distributing, combining like terms, and isolating y to find its value.
Substitute y back into the second equation to solve for x: Now that we have found the value of y, we can substitute it back into the second equation to solve for x:
```
x = 18 + 9
x = 27
```
This substitution replaces y in the second equation with its value, allowing us to calculate the value of x.

Therefore, the solution to the system of equations is x = 27 and y = 18. This means that the store rented 27 pairs of skis and 18 snowboards on that day.

To ensure the accuracy of our solution, it's crucial to verify that the values we obtained for x and y satisfy both equations in the system. This step helps identify any potential errors in our calculations and provides confidence in the correctness of our answer.

Substitute x = 27 and y = 18 into the first equation:
```
44(27) + 58(18) = 2232
1188 + 1044 = 2232
2232 = 2232
```
The equation holds true, indicating that the values satisfy the first condition.
Substitute x = 27 and y = 18 into the second equation:
```
27 = 18 + 9
27 = 27
```
The equation also holds true, confirming that the values satisfy the second condition.

Since the values x = 27 and y = 18 satisfy both equations in the system, we can confidently conclude that our solution is correct. The store rented 27 pairs of skis and 18 snowboards on that day.

In this article, we have explored the application of systems of equations to a real-world scenario involving ski and snowboard rentals. By formulating and solving a system of equations, we were able to determine the number of skis and snowboards rented on a particular day. This example highlights the power and versatility of systems of equations in modeling and solving problems across various domains. By mastering this fundamental mathematical concept, readers can enhance their problem-solving skills and gain valuable insights into real-world applications. The ability to translate real-world scenarios into mathematical models and solve them effectively is a crucial skill in various fields, from business and finance to engineering and science.

Systems of equations
Ski rentals
Snowboard rentals
Substitution method
Problem-solving
Mathematical modeling