the table shows the total cost of purchasing x same - priced items and a catalog. what is the initial value…

Explanation:

Step1: Find the cost per item

The cost per item is the slope. Using two points ((1,10)) and ((2,14)), the slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{14 - 10}{2 - 1}=4).

Step2: Find the initial value (y - intercept)

Using the point - slope form (y - y_1=m(x - x_1)) with (m = 4) and ((x_1,y_1)=(1,10)), we have (y-10 = 4(x - 1)). Expanding gives (y=4x+6). When (x = 0) (initial value, before any items are bought), (y = 6) does not make sense. Wait, another approach: The total cost formula is (y=mx + b). We know (m = 4) (from step1). Substitute (x = 1,y = 10) into (y=4x + b), (10=4\times1 + b), so (b = 6). But when (x = 0) (no items bought), (y=b). But in the context of the problem, the total cost is of purchasing items and a catalog. The equation of the line is (y = 4x+6). The (y) - intercept (when (x = 0)) is not in the table. But if we consider the cost model (y=\text{cost per item}\times x+\text{cost of catalog}). We found cost per item (m = 4). Then from (y=4x + b) and using (x = 1,y = 10), (10=4\times1+b), (b = 6). The initial value (when (x = 0), but in the context of the problem, we assume the catalog is always bought. The cost model is (y) (total cost)= (cost per item)(\times x+) (cost of catalog).

Answer:

($6), the cost of the catalog.