To determine which equation models Hanna's purchases of socks and blouses, let's carefully define and understand the given variables and their relationships.
1. Cost of each pair of socks: \[tex]$2.99
2. Cost of each blouse: \$[/tex]12.99
3. Total amount spent: \$43.92
4. Number of pairs of socks purchased: [tex]\(x\)[/tex]
5. Number of blouses purchased: [tex]\(y\)[/tex]
The goal is to find an equation that relates the number of socks and blouses Hanna bought with the total amount spent.
Given any shopping scenario where different items are purchased, the total cost is calculated by multiplying the number of each item by its respective price and then summing these products. Therefore, the total cost equation for Hanna's purchases can be expressed as:
[tex]\[ \text{(Cost per pair of socks)} \times \text{(Number of pairs of socks)} + \text{(Cost per blouse)} \times \text{(Number of blouses)} = \text{Total cost} \][/tex]
Substituting the given values into this equation:
[tex]\[ 2.99x + 12.99y = 43.92 \][/tex]
Now let's examine each provided option to identify the correct one:
1. [tex]\(x + y = 15.98\)[/tex]
- This option does not relate the total cost to the individual prices of socks and blouses. It also introduces an unrelated total (15.98), which does not help us model the total expenditure.
2. [tex]\(x + y = 43.92\)[/tex]
- This option makes an incorrect assumption that the number of items adds up to the total cost, which does not match our situation where prices are different.
3. [tex]\(43.92x - 2.99y = 12.99\)[/tex]
- This equation incorrectly mixes the terms and operations, not matching the way to combine the costs of multiple items for a total figure.
4. [tex]\(2.99x + 12.99y = 43.92\)[/tex]
- This is the correct equation. It accurately represents the total expenditure by summing up the products of the quantities and their respective prices.
Therefore, the correct equation that models Hanna's purchases is:
[tex]\[ \boxed{2.99x + 12.99y = 43.92} \][/tex]
