Answer :
To determine the initial value and what it represents, we can analyze the provided table of costs. The table given:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Number of Items } (x) & \text{Total Cost } (y) \\ \hline 1 & \$10 \\ \hline 2 & \$14 \\ \hline 3 & \$18 \\ \hline 4 & \$22 \\ \hline \end{tabular} \][/tex]
includes the number of items and the corresponding total cost, which comprises the cost of the items plus the cost of a catalog.
### Step 1: Determine the Cost Per Item
First, let's calculate the cost per item. We compare the total cost for different numbers of items to see the increment.
[tex]\[ \text{Cost of 2 items} - \text{Cost of 1 item} = \[tex]$14 - \$[/tex]10 = \$4
\][/tex]
[tex]\[ \text{Cost of 3 items} - \text{Cost of 2 items} = \[tex]$18 - \$[/tex]14 = \$4
\][/tex]
[tex]\[ \text{Cost of 4 items} - \text{Cost of 3 items} = \[tex]$22 - \$[/tex]18 = \$4
\][/tex]
From these calculations, the cost per item is consistent and equals \$4.
### Step 2: Determine the Cost of the Catalog
We now focus on the total cost expression:
[tex]\[ \text{Total Cost} = \text{Cost of Catalog} + (\text{Cost Per Item} \times \text{Number of Items}) \][/tex]
Using the information from the table and the cost per item (\$4), we use the first total cost entry to find the catalog cost when 1 item is purchased:
[tex]\[ \[tex]$10 = \text{Cost of Catalog} + (\$[/tex]4 \times 1)
\][/tex]
[tex]\[ \[tex]$10 = \text{Cost of Catalog} + \$[/tex]4
\][/tex]
[tex]\[ \text{Cost of Catalog} = \[tex]$10 - \$[/tex]4 = \$6
\][/tex]
### Step 3: Conclusion
Based on our calculations:
- The cost per item is \$4.
- The cost of the catalog is \$6.
Thus, the initial value \( \$6 \) represents the cost of the catalog. So among the given options, the correct identification is:
[tex]\[ \$6, \text{ the cost of the catalog} \][/tex]
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Number of Items } (x) & \text{Total Cost } (y) \\ \hline 1 & \$10 \\ \hline 2 & \$14 \\ \hline 3 & \$18 \\ \hline 4 & \$22 \\ \hline \end{tabular} \][/tex]
includes the number of items and the corresponding total cost, which comprises the cost of the items plus the cost of a catalog.
### Step 1: Determine the Cost Per Item
First, let's calculate the cost per item. We compare the total cost for different numbers of items to see the increment.
[tex]\[ \text{Cost of 2 items} - \text{Cost of 1 item} = \[tex]$14 - \$[/tex]10 = \$4
\][/tex]
[tex]\[ \text{Cost of 3 items} - \text{Cost of 2 items} = \[tex]$18 - \$[/tex]14 = \$4
\][/tex]
[tex]\[ \text{Cost of 4 items} - \text{Cost of 3 items} = \[tex]$22 - \$[/tex]18 = \$4
\][/tex]
From these calculations, the cost per item is consistent and equals \$4.
### Step 2: Determine the Cost of the Catalog
We now focus on the total cost expression:
[tex]\[ \text{Total Cost} = \text{Cost of Catalog} + (\text{Cost Per Item} \times \text{Number of Items}) \][/tex]
Using the information from the table and the cost per item (\$4), we use the first total cost entry to find the catalog cost when 1 item is purchased:
[tex]\[ \[tex]$10 = \text{Cost of Catalog} + (\$[/tex]4 \times 1)
\][/tex]
[tex]\[ \[tex]$10 = \text{Cost of Catalog} + \$[/tex]4
\][/tex]
[tex]\[ \text{Cost of Catalog} = \[tex]$10 - \$[/tex]4 = \$6
\][/tex]
### Step 3: Conclusion
Based on our calculations:
- The cost per item is \$4.
- The cost of the catalog is \$6.
Thus, the initial value \( \$6 \) represents the cost of the catalog. So among the given options, the correct identification is:
[tex]\[ \$6, \text{ the cost of the catalog} \][/tex]
