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. Which

Introduction

In the world of mathematics, real-life scenarios often serve as the perfect backdrop for exploring various concepts and principles. Hanna's shopping spree is a great example of this, as it allows us to delve into the realm of algebra and graphing. In this article, we will explore the mathematical aspects of Hanna's sock and blouse purchases, examining the relationships between the number of pairs of socks and blouses she buys.

The Cost of Socks and Blouses

Hanna is shopping for two items: socks and blouses. Each pair of socks costs $\$2.99$, while each blouse costs $\$12.99$. We can represent the cost of each item using the following equations:

• Cost of socks: $2.99x$
• Cost of blouses: $12.99y$

where $x$ represents the number of pairs of socks purchased and $y$ represents the number of blouses purchased.

The Total Cost of Hanna's Purchase

The total cost of Hanna's purchase is the sum of the cost of the socks and the cost of the blouses. We can represent this using the following equation:

Total Cost = Cost of Socks + Cost of Blouses = $2.99x + 12.99y$

Graphing the Total Cost

To visualize the relationship between the number of pairs of socks and blouses purchased, we can graph the total cost equation. We can use a coordinate plane, with the x-axis representing the number of pairs of socks purchased and the y-axis representing the number of blouses purchased.

Graphing the Cost of Socks

First, let's graph the cost of socks equation: $2.99x$. This is a linear equation, and its graph is a straight line with a slope of 2.99.

Graphing the Cost of Blouses

Next, let's graph the cost of blouses equation: $12.99y$. This is also a linear equation, and its graph is a straight line with a slope of 12.99.

Graphing the Total Cost

Now, let's graph the total cost equation: $2.99x + 12.99y$. This is a linear equation, and its graph is a straight line with a slope of 2.99 + 12.99 = 15.98.

Interpreting the Graph

The graph of the total cost equation represents the relationship between the number of pairs of socks and blouses purchased and the total cost of Hanna's purchase. We can use this graph to answer questions such as:

• What is the total cost of purchasing 5 pairs of socks and 2 blouses?
• How many pairs of socks and blouses must Hanna purchase to spend a total of $\$100$?

Solving Systems of Equations

In addition to graphing, we can also solve systems of equations to find the number of pairs of socks and blouses purchased. A system of equations is a set of two or more equations that are solved simultaneously.

Example 1: Solving a System of Equations

Suppose Hanna purchases 5 pairs of socks and 2 blouses. We can represent this using the following system of equations:

• $2.99x = 14.95$
• $12.99y = 25.98$

We can solve this system of equations by substituting the value of $x$ into the second equation:

$12.99y = 25.98$ $y = \frac{25.98}{12.99}$ $y = 2$

Therefore, Hanna purchases 2 blouses.

Example 2: Solving a System of Equations

Suppose Hanna spends a total of $\$100$ on socks and blouses. We can represent this using the following system of equations:

• $2.99x + 12.99y = 100$
• $x = 5$

We can solve this system of equations by substituting the value of $x$ into the first equation:

$2.99(5) + 12.99y = 100$ $14.95 + 12.99y = 100$ $12.99y = 85.05$ $y = \frac{85.05}{12.99}$ $y = 6.55$

Therefore, Hanna purchases 6.55 blouses.

Conclusion

In conclusion, Hanna's shopping spree is a great example of how real-life scenarios can be used to explore various mathematical concepts and principles. By graphing the total cost equation and solving systems of equations, we can gain a deeper understanding of the relationships between the number of pairs of socks and blouses purchased and the total cost of Hanna's purchase.