To determine the equation that represents the total cost [tex]\( y \)[/tex] of leasing a car for [tex]\( x \)[/tex] months, we need to find a linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
The general form of the equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
### Step 1: Calculate the Slope (m)
The slope [tex]\( m \)[/tex] is calculated using two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] from the table. We will use the points [tex]\((1, 1859)\)[/tex] and [tex]\((12, 5808)\)[/tex].
Using the formula for the slope:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the values from the selected points:
[tex]\[ m = \frac{5808 - 1859}{12 - 1} \][/tex]
[tex]\[ m = \frac{3949}{11} \][/tex]
[tex]\[ m = 359 \][/tex]
### Step 2: Calculate the Y-Intercept (b)
The y-intercept [tex]\( b \)[/tex] is found by using one of the points and the slope we just calculated. Let's use the point [tex]\((1, 1859)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
Plugging in the values:
[tex]\[ 1859 = 359 \cdot 1 + b \][/tex]
[tex]\[ 1859 = 359 + b \][/tex]
[tex]\[ b = 1859 - 359 \][/tex]
[tex]\[ b = 1500 \][/tex]
### Step 3: Write the Equation in Slope-Intercept Form
Now that we have the slope [tex]\( m = 359 \)[/tex] and the y-intercept [tex]\( b = 1500 \)[/tex], we can write the equation:
[tex]\[ y = 359x + 1500 \][/tex]
So, the equation that represents the total cost [tex]\( y \)[/tex] of leasing a car for [tex]\( x \)[/tex] months is:
[tex]\[ y = 359x + 1500 \][/tex]
