To find the initial value and understand what it represents, let us analyze the given data step by step.
We are given the table of the total cost \( y \) for purchasing \( x \) same-priced items along with the catalog:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of Items } (x) & \text{Total Cost } (y) \\
\hline
1 & \$10 \\
\hline
2 & \$14 \\
\hline
3 & \$18 \\
\hline
4 & \$22 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Identify the relationship:
The total cost \( y \) is a linear function of the number of items \( x \). We can represent it in the form:
[tex]\[
y = mx + b
\][/tex]
where:
- \( m \) is the slope (cost per item),
- \( b \) is the y-intercept (initial value, representing the cost of the catalog).
2. Calculate the slope \( m \):
The slope \( m \) can be calculated using any two points from the table. For instance, using the points \((1, 10)\) and \((2, 14)\):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the values:
[tex]\[
m = \frac{14 - 10}{2 - 1} = \frac{4}{1} = 4
\][/tex]
Thus, the cost per item is \( \boldsymbol{\$4} \).
3. Find the y-intercept \( b \):
Using one of the points and the value of \( m \), we can solve for \( b \). Let's use the point \((1, 10)\):
[tex]\[
y = mx + b
\][/tex]
Substituting \( y = 10 \), \( x = 1 \), and \( m = 4 \):
[tex]\[
10 = 4 \cdot 1 + b \\
10 = 4 + b \\
b = 10 - 4 = 6
\][/tex]
Therefore, the initial value \( b \) is \( \boldsymbol{\$6} \).
### Interpretation:
- \$4 represents the cost per item.
- \$6 represents the cost of the catalog.
So, the correct answers are:
- \(\$4\), the cost per item
- [tex]\(\$6\)[/tex], the cost of the catalog
