Certainly! To write an equation in slope-intercept form to represent the total cost [tex]\( y \)[/tex] of leasing a car for [tex]\( x \)[/tex] months, we need to find the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) of the line described by the given data.
### Step-by-Step Solution:
1. Identify the given points:
The table provides us with the following data points:
- (1, 1859)
- (3, 2577)
- (8, 4372)
- (12, 5808)
2. Calculate the slope ([tex]\( m \)[/tex]):
The slope ([tex]\( m \)[/tex]) is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
We'll use the first data point (1, 1859) and the last data point (12, 5808) to find the slope:
[tex]\[
m = \frac{5808 - 1859}{12 - 1} = \frac{3949}{11} = 359
\][/tex]
3. Calculate the y-intercept ([tex]\( b \)[/tex]):
The y-intercept ([tex]\( b \)[/tex]) can be found using the slope [tex]\( m \)[/tex] and any point on the line. Using the first data point (1, 1859):
[tex]\[
y = mx + b
\][/tex]
Substituting [tex]\( y = 1859 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( m = 359 \)[/tex]:
[tex]\[
1859 = 359 \cdot 1 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 1859 - 359 = 1500
\][/tex]
4. Write the equation in slope-intercept form:
The slope-intercept form of a line is given by [tex]\( y = mx + b \)[/tex]. So, substituting [tex]\( m \)[/tex] and [tex]\( b \)[/tex] we get:
[tex]\[
y = 359x + 1500
\][/tex]
### Final Answer
The equation in slope-intercept form that represents the total cost [tex]\( y \)[/tex] of leasing a car for [tex]\( x \)[/tex] months is:
[tex]\[
y = 359x + 1500
\][/tex]
