To determine which linear equation represents the situation accurately, we need to find the linear relationship between the miles traveled ([tex]$x$[/tex]) and the cost ([tex]$y$[/tex]).
A linear equation typically takes the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope (rate of change) and [tex]\( b \)[/tex] is the y-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]).
### Step-by-Step Solution:
1. Identify two data points from the given table:
- The first point is (2, 8.50)
- The second point is (5, 15.25)
2. Calculate the slope [tex]\( m \)[/tex]:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values,
[tex]\[ m = \frac{15.25 - 8.50}{5 - 2} = \frac{6.75}{3} = 2.25 \][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
We know the equation of the line is [tex]\( y = mx + b \)[/tex]. To find [tex]\( b \)[/tex], we can use either of the points. Let’s use the point (2, 8.50):
[tex]\[ y = mx + b \implies 8.50 = 2.25 \cdot 2 + b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ 8.50 = 4.50 + b \][/tex]
[tex]\[ b = 8.50 - 4.50 = 4.00 \][/tex]
4. Form the equation:
Substituting [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the linear equation form:
[tex]\[ y = 2.25x + 4.00 \][/tex]
### Conclusion:
The linear equation that best represents the situation is:
[tex]\[ y = 2.25x + 4.00 \][/tex]
Therefore, the correct equation from the given choices is:
[tex]\[ \boxed{2} \][/tex]
