Mathematics

# 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?

y - value x - year $24000=a\cdot3+b\\ 19000=a\cdot5+b\\\\ 24000=3a+b\\ -19000=-5a-b\\ ---------\\ 5000=-2a\\ a=-2500\\\\ b+3\cdot(-2500)=24000\\ b-7500=24000\\ b=31500\\\\ \boxed{y=-2500x+31500}$ a) $f(x)=-2500x+31500\\ f(x+1)=-2500(x+1)+31500\\ f(x+1)=-2500x-2500+31500\\ f(x+1)=-2500x+29000\\\\ f(x)-f(x+1)=-2500x+31500-(-2500x+29000)\\ f(x)-f(x+1)=-2500x+31500+2500x-29000\\ f(x)-f(x+1)=\boxed{2500}$ b) $f(0)=-2500\cdot0+31500=\boxed{31500}$ c) $f(10)=-2500\cdot10+31500\\ f(10)=-25000+31500\\ f(10)=\boxed{6500}$