Three years after purchase, a car is estimated to be worth $24,000. At five years, its value is $19,000. If the car is depreciating in a linear manner, write an equation that represents the depreciation of the car. Answer the following questions:

a. How much is the car depreciating each year?
b. What was the purchase price of the car?
c. If the car continues this rate of depreciation, what will its value be at year 10?

To represent the depreciation of the car in a linear manner, we can use the equation for a straight line, which is in the form of y = mx + b. In this case, let x represent the number of years since purchase, and y represent the value of the car in dollars.

We are given two points on the line:
(3, $24,000) and (5, $19,000).

We can use these two points to find the slope of the line (m), and then use the slope and one of the points to find the y-intercept (b), which represents the initial value (purchase price) of the car.

To find the slope (m), we use the formula:
m = (y2 - y1) / (x2 - x1)

Using the points (3, $24,000) and (5, $19,000):
m = ($19,000 - $24,000) / (5 - 3)
m = -$5,000 / 2
m = -$2,500

The slope of the line is -2,500.

Now, we can use the slope and one of the points (3, $24,000) to find the y-intercept (b).

Using the point-slope form of a linear equation:
y - y1 = m(x - x1)

Using (3, $24,000):
y - $24,000 = -$2,500(x - 3)
y - $24,000 = -$2,500x + $7,500
y = -$2,500x + $7,500 + $24,000
y = -$2,500x + $31,500

Therefore, the equation that represents the depreciation of the car is:
y = -$2,500x + $31,500

a. How much is the car depreciating each year?
The car is depreciating at a rate of $2,500 per year.

b. What was the purchase price of the car?
The purchase price of the car is represented by the y-intercept, which is $31,500.

c. If the car continues this rate of depreciation, what will its value be at year 10?
To find the value of the car at year 10 (x = 10), we substitute x = 10 into the equation:
y = -$2,500(10) + $31,500
y = -$25,000 + $31,500
y = $6,500

Therefore, the value of the car at year 10 would be $6,500.

To write the equation that represents the depreciation of the car, we can use the slope-intercept form of a linear equation: y = mx + b, where y represents the value of the car in dollars, x represents the number of years, m represents the depreciation rate (depreciation per year), and b represents the initial value (purchase price) of the car.

We are given the values at three points: (3, 24000) represents the value of the car after 3 years, (5, 19000) represents the value of the car after 5 years, and the current year (0) represents the purchase price of the car.

a) To find the depreciation rate (m), we can use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) = (3, 24000) and (x2, y2) = (5, 19000).

m = (19000 - 24000) / (5 - 3)
m = -5000 / 2
m = -2500

Therefore, the car is depreciating $2500 each year.

b) To find the purchase price of the car, we can substitute the known value of the car at year 0 into the equation.

0 = (-2500 * 0) + b
b = 0

Therefore, the purchase price of the car was $0 (assuming it was given or is a typo).

c) To find the value of the car at year 10, we can substitute the value of x = 10 into the equation.

y = -2500 * 10 + b
y = -25000 + b

Since the value of b is not given, we can't find the exact value of the car at year 10 without knowing the purchase price.

value = v

purchase price = p

v = p - k t

24,000 = p - 3 k
19,000 = p - 5 k
------------------ subtract
5,000 = 2 k
k = 2,500 per year (part a)

24,000 = p - 7,500
p = 31,500

v(10) = 31,500 - 25,000 = 6,500