The value of a car decreases at a constant rate. After 1 year the value of the car is $20,000. After 2 more yearsit is $14,000. Write an equation in slope-intercept that represents the value y (in dollars) of the car after x years. What is the y-intercept of the line? In this problem, what does the y-intercept represent?
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To find the equation in slope-intercept form that represents the value of the car over time, we need to determine the slope and y-intercept.
First, let's identify the given information:
After 1 year, the value of the car is $20,000.
After 2 more years (a total of 3 years), the value is $14,000.
Using this information, we can find the slope (m) of the line using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (1, $20,000) and (x2, y2) = (3, $14,000).
m = (14000 - 20000) / (3 - 1)
m = -6000 / 2
m = -3000
Now, the slope-intercept form of a linear equation is given by:
y = mx + b
where m represents the slope and b represents the y-intercept.
Using the slope (m = -3000), we can substitute it into the equation:
y = -3000x + b
To find the y-intercept (b), we can substitute the value of x and y from one of the given points. Let's use (1, $20,000):
20000 = -3000(1) + b
20000 = -3000 + b
b = 23000
Therefore, the equation that represents the value y (in dollars) of the car after x years is:
y = -3000x + 23000
The y-intercept of the line is 23000. In this context, the y-intercept represents the initial value of the car when x (years) is equal to zero. So, it represents the starting value or the original price of the car.