Proof of the Elimination Method

Oct 13, 2025 by miodeya.com 43 views

Hanna's Shopping Spree: A Mathematical Exploration of Sock and Blouse Purchases

In the world of mathematics, real-life scenarios can be used to create engaging and interactive problems. One such scenario is Hanna's shopping spree, where she purchases socks and blouses at a local store. In this article, we will delve into the mathematical world of Hanna's shopping spree, exploring the relationships between the number of pairs of socks and blouses she purchases.

Hanna shops for socks that cost $\$2.99$ for each pair and blouses that cost $\$12.99$ each. Let $x$ represent the number of pairs of socks purchased, and let $y$ represent the number of blouses purchased. The total cost of the socks and blouses can be represented by the equation:

2.99x + 12.99y = C

where $C$ is the total cost of the socks and blouses.

In this problem, we have two variables: $x$ and $y$ . The variable $x$ represents the number of pairs of socks purchased, while the variable $y$ represents the number of blouses purchased. We can think of these variables as the inputs to the problem, and the total cost $C$ as the output.

The cost of each pair of socks is $\$2.99$ . This means that for every pair of socks Hanna purchases, she will pay $\$2.99$ . If she purchases $x$ pairs of socks, the total cost of the socks will be:

2.99x

The cost of each blouse is $\$12.99$ . This means that for every blouse Hanna purchases, she will pay $\$12.99$ . If she purchases $y$ blouses, the total cost of the blouses will be:

12.99y

The total cost of the socks and blouses is the sum of the cost of the socks and the cost of the blouses. This can be represented by the equation:

2.99x + 12.99y = C

where $C$ is the total cost of the socks and blouses.

To solve for $x$ and $y$ , we need to find the values of these variables that satisfy the equation:

2.99x + 12.99y = C

We can do this by using algebraic methods, such as substitution or elimination.

Substitution Method

One way to solve for $x$ and $y$ is to use the substitution method. We can solve for $x$ in terms of $y$ by isolating $x$ in the equation:

2.99x + 12.99y = C

2.99x = C - 12.99y

x = \frac{C - 12.99y}{2.99}

Elimination Method

Another way to solve for $x$ and $y$ is to use the elimination method. We can multiply the first equation by $2.99$ and the second equation by $12.99$ to eliminate the variable $x$ :

2.99(2.99x + 12.99y) = 2.99C

12.99(2.99x + 12.99y) = 12.99C

8.9801x + 39.0047y = 2.99C

38.0047x + 159.987y = 12.99C

Subtracting the first equation from the second equation, we get:

29.0246x + 120.983y = 10.99C

Now that we have eliminated the variable $x$ , we can solve for $y$ by isolating $y$ in the equation:

29.0246x + 120.983y = 10.99C

120.983y = 10.99C - 29.0246x

y = \frac{10.99C - 29.0246x}{120.983}

In this article, we have explored the mathematical world of Hanna's shopping spree, where she purchases socks and blouses at a local store. We have used algebraic methods, such as substitution and elimination, to solve for the number of pairs of socks and blouses purchased. The total cost of the socks and blouses can be represented by the equation:

2.99x + 12.99y = C

where $C$ is the total cost of the socks and blouses.

This problem has real-world applications in the field of economics and finance. For example, a business owner may use this type of problem to determine the optimal number of products to produce and sell, given the costs of production and the demand for the products.

Future research directions in this area may include:

Developing more complex models that take into account additional variables, such as the cost of labor and the demand for the products.
Using numerical methods, such as the Newton-Raphson method, to solve for the number of pairs of socks and blouses purchased.
Exploring the use of this type of problem in other fields, such as marketing and management.

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Economics" by Gregory Mankiw

A. Proof of the Elimination Method

To prove the elimination method, we can start with the two equations:

2.99x + 12.99y = C

38.0047x + 159.987y = 12.99C

We can multiply the first equation by $2.99$ and the second equation by $12.99$ to eliminate the variable $x$ :

2.99(2.99x + 12.99y) = 2.99C

12.99(2.99x + 12.99y) = 12.99C

8.9801x + 39.0047y = 2.99C

38.0047x + 159.987y = 12.99C

Subtracting the first equation from the second equation, we get:

29.0246x + 120.983y = 10.99C

B. Proof of the Substitution Method

To prove the substitution method, we can start with the two equations:

2.99x + 12.99y = C

x = \frac{C - 12.99y}{2.99}

We can substitute the expression for $x$ into the first equation:

2.99\left(\frac{C - 12.99y}{2.99}\right) + 12.99y = C

Simplifying the equation, we get:

C - 12.99y + 12.99y = C

C = C

This shows that the substitution method is a valid method for solving for $x$ and $y$ .
Hanna's Shopping Spree: A Mathematical Exploration of Sock and Blouse Purchases - Q&A

In our previous article, we explored the mathematical world of Hanna's shopping spree, where she purchases socks and blouses at a local store. We used algebraic methods, such as substitution and elimination, to solve for the number of pairs of socks and blouses purchased. In this article, we will answer some of the most frequently asked questions about Hanna's shopping spree.

Q: What is the total cost of the socks and blouses?

A: The total cost of the socks and blouses is represented by the equation:

2.99x + 12.99y = C

where $C$ is the total cost of the socks and blouses.

Q: How do I solve for $x$ and $y$ ?

A: To solve for $x$ and $y$ , you can use algebraic methods, such as substitution or elimination. We have provided a step-by-step guide on how to use these methods in our previous article.

Q: What is the cost of each pair of socks?

A: The cost of each pair of socks is $\$2.99$ .

Q: What is the cost of each blouse?

A: The cost of each blouse is $\$12.99$ .

Q: How do I determine the optimal number of pairs of socks and blouses to purchase?

A: To determine the optimal number of pairs of socks and blouses to purchase, you can use the equation:

2.99x + 12.99y = C

where $C$ is the total cost of the socks and blouses. You can also use numerical methods, such as the Newton-Raphson method, to solve for the optimal number of pairs of socks and blouses.

Q: Can I use this type of problem in other fields, such as marketing and management?

A: Yes, you can use this type of problem in other fields, such as marketing and management. For example, a business owner may use this type of problem to determine the optimal number of products to produce and sell, given the costs of production and the demand for the products.

Q: What are some real-world applications of this type of problem?

A: Some real-world applications of this type of problem include:

Determining the optimal number of products to produce and sell, given the costs of production and the demand for the products.
Calculating the total cost of a project, given the costs of individual components and the number of components required.
Determining the optimal number of employees to hire, given the costs of hiring and the demand for the product or service.

In this article, we have answered some of the most frequently asked questions about Hanna's shopping spree. We have also provided a step-by-step guide on how to use algebraic methods, such as substitution and elimination, to solve for the number of pairs of socks and blouses purchased. We hope that this article has been helpful in understanding the mathematical world of Hanna's shopping spree.

This type of problem has real-world applications in many fields, including:

Economics: Determining the optimal number of products to produce and sell, given the costs of production and the demand for the products.
Finance: Calculating the total cost of a project, given the costs of individual components and the number of components required.
Marketing: Determining the optimal number of employees to hire, given the costs of hiring and the demand for the product or service.
Management: Determining the optimal number of resources to allocate, given the costs of resources and the demand for the product or service.

Future research directions in this area may include:

Developing more complex models that take into account additional variables, such as the cost of labor and the demand for the products.
Using numerical methods, such as the Newton-Raphson method, to solve for the number of pairs of socks and blouses purchased.
Exploring the use of this type of problem in other fields, such as marketing and management.

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Economics" by Gregory Mankiw

A. Proof of the Elimination Method

To prove the elimination method, we can start with the two equations:

2.99x + 12.99y = C

38.0047x + 159.987y = 12.99C

We can multiply the first equation by $2.99$ and the second equation by $12.99$ to eliminate the variable $x$ :

2.99(2.99x + 12.99y) = 2.99C

12.99(2.99x + 12.99y) = 12.99C

8.9801x + 39.0047y = 2.99C

38.0047x + 159.987y = 12.99C

Subtracting the first equation from the second equation, we get:

29.0246x + 120.983y = 10.99C

B. Proof of the Substitution Method

To prove the substitution method, we can start with the two equations:

2.99x + 12.99y = C

x = \frac{C - 12.99y}{2.99}

We can substitute the expression for $x$ into the first equation:

2.99\left(\frac{C - 12.99y}{2.99}\right) + 12.99y = C

Simplifying the equation, we get:

C - 12.99y + 12.99y = C

C = C

This shows that the substitution method is a valid method for solving for $x$ and $y$ .