To determine the initial value and what it represents, let's analyze the given table:
[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]
1. Determine the cost per item:
The change in total cost when purchasing each additional item is consistent:
- From 1 to 2 items: [tex]\( \$ 14 - \$ 10 = \$ 4 \)[/tex]
- From 2 to 3 items: [tex]\( \$ 18 - \$ 14 = \$ 4 \)[/tex]
- From 3 to 4 items: [tex]\( \$ 22 - \$ 18 = \$ 4 \)[/tex]
Therefore, the cost per item is [tex]\( \$ 4 \)[/tex].
2. Determine the initial value:
The initial value is the total cost when [tex]\( x = 0 \)[/tex] (before any items are purchased). The equation representing the total cost [tex]\( y \)[/tex] is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the cost per item and [tex]\( b \)[/tex] is the initial value.
Using the point [tex]\((1, 10)\)[/tex]:
[tex]\[
y = mx + b \implies 10 = 4 \cdot 1 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
10 = 4 + b \implies b = 10 - 4 = 6
\][/tex]
Conclusion:
- The cost per item is [tex]\( \$ 4 \)[/tex].
- The initial value is [tex]\( \$ 6 \)[/tex].
The initial value [tex]\( \$ 6 \)[/tex] represents the cost of the catalog.
So from the given options:
- [tex]\( \$ 4 \)[/tex], the cost per item
- [tex]\( \$ 6 \)[/tex], the cost of the catalog
Correct answers:
- [tex]\( \$ 4 \)[/tex], the cost per item
- [tex]\( \$ 6 \)[/tex], the cost of the catalog
