To determine which scenario matches the given equation [tex]\( y - 53.75 = 3.25(x - 15) \)[/tex], we need to analyze the parts of the equation and match them to the given scenarios. Here's the step-by-step process:
### Step 1: Standard Form Comparison
The given equation is:
[tex]\[ y - 53.75 = 3.25(x - 15) \][/tex]
Let's rewrite this equation in the standard linear form [tex]\( y = mx + c \)[/tex]:
[tex]\[ y - 53.75 = 3.25(x - 15) \][/tex]
Add [tex]\( 53.75 \)[/tex] to both sides:
[tex]\[ y = 3.25(x - 15) + 53.75 \][/tex]
Distribute the [tex]\( 3.25 \)[/tex]:
[tex]\[ y = 3.25x - 48.75 + 53.75 \][/tex]
Combine like terms:
[tex]\[ y = 3.25x + 5 \][/tex]
Now, we have the equation in the [tex]\( y = mx + c \)[/tex] form, where:
- [tex]\( m = 3.25 \)[/tex], which represents the slope or the cost per widget.
- [tex]\( c = 5 \)[/tex], which represents the y-intercept or the flat shipping rate.
### Step 2: Scenario Analysis
Let's compare each scenario with the components of the equation:
#### 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.
In this scenario:
- The cost per widget = \[tex]$3.25 (matches the slope, \( m \))
- There is a flat shipping rate that affects the total cost (matches the y-intercept, \( c \))
#### 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.
In this scenario:
- A flat shipping rate of \$[/tex]3.25 is mentioned, but it should match the cost per widget, not the shipping rate.
- This does not match our interpretation of slope and intercept from the equation.
#### 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.
In this scenario:
- The cost per widget is \$[/tex]15, which doesn't match our slope [tex]\( m = 3.25 \)[/tex].
Therefore, the only scenario that matches our equation where the flat shipping rate is the y-intercept (\[tex]$5) and the cost per widget is the slope (\$[/tex]3.25) is Scenario 1.
### Conclusion
The correct scenario is:
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.
Thus, the answer is Scenario 1.
