To match the given equation [tex]$y - 53.75 = 3.25(x - 15)$[/tex], we need to analyze each scenario by converting them into a form that can fit into the linear equation representation. Let's break down each scenario step by step.
### Scenario 1:
A manufacturer sells widgets for [tex]$3.25 each. They charge a flat shipping rate on all orders. An order of 15 widgets costs $[/tex]53.75.
1. Let the number of widgets be [tex]\( x \)[/tex] and the total cost be [tex]\( y \)[/tex].
2. The cost of widgets would be [tex]$3.25$[/tex] each, so the cost for [tex]\( x \)[/tex] widgets is [tex]\( 3.25x \)[/tex].
3. There's a flat shipping rate (let's call it [tex]\( r \)[/tex]). So, the total cost [tex]\( y \)[/tex] is given by:
[tex]\[
y = 3.25x + r
\][/tex]
4. We know that when [tex]\( x = 15 \)[/tex], [tex]\( y = 53.75 \)[/tex]:
[tex]\[
53.75 = 3.25 \times 15 + r
\][/tex]
5. Calculate:
[tex]\[
53.75 = 48.75 + r
\][/tex]
6. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = 53.75 - 48.75 = 5
\][/tex]
7. Thus, the equation becomes:
[tex]\[
y = 3.25x + 5
\][/tex]
This matches the given equation when rearranged: [tex]\( y - 53.75 = 3.25(x - 15) \)[/tex].
### Scenario 2:
A manufacturer sells widgets. They charge a flat shipping rate of [tex]$3.25 on all orders. An order of 15 widgets costs $[/tex]53.75.
1. There is a set flat shipping rate of [tex]$3.25, so we include this in the final cost.
2. Denote the price per widget as \( p \). The total cost \( y \) for \( x \) widgets then becomes:
\[
y = px + 3.25
\]
3. Given \( x = 15 \), \( y = 53.75 \):
\[
53.75 = p \times 15 + 3.25
\]
4. Rearrange and solve for \( p \):
\[
53.75 - 3.25 = p \times 15
\]
\[
50.50 = p \times 15
\]
\[
p \approx 3.37
\]
5. Therefore, the equation would be approximated to:
\[
y \approx 3.37x + 3.25
\]
This does not match the given equation \( y - 53.75 = 3.25(x - 15) \).
### Scenario 3:
A manufacturer sells widgets for $[/tex]15 each. They charge a flat shipping rate of [tex]$3.25 on all orders. An order costs $[/tex]53.75.
1. Using the price per widget of [tex]$15:
\[
y = 15x + 3.25
\]
2. Given \( x = 1 \) (only one widget price can reach the total cost $[/tex]53.75 due to high price per widget of [tex]$15):
\[
15 \times 1 + 3.25 = 18.25,
\]
which is far from $[/tex]53.75. Hence, this scenario does not fit.
### Conclusion:
The scenario that correctly matches the equation [tex]$y - 53.75 = 3.25(x - 15)$[/tex] is:
Scenario 1: A manufacturer sells widgets for [tex]$3.25 each. They charge a flat shipping rate on all orders. An order of 15 widgets costs $[/tex]53.75.
