Acar manufacturer is reducing the number of incidents with the transmission by issuing a voluntary recall. during week 10 of the recall, the manufacturer fixed 200 cars. in week 15, the manufacturer fixed 175 cars. assume that the reduction in the number of cars each week is linear. write an equation in function form to show the number of cars seen each week by the mechanic

f(x) = -5x + 250

Step-by-step explanation:

* Lets  explain how to solve the problem

- In week 10 the manufacturer fixed 200 cars

- In week 15, the manufacturer fixed 175 cars

- the reduction in the number of cars each week is linear

- The form of the linear equation is y = mx + c, where m is the slope of

 the line which represent the equation and c is the y-intercept

- The slope of the line m =

  where (x1 , y1) and (x2 , y2) are two points on the line

* Lets solve the problem

- Assume that the weeks' number is x and the cars' number is y

∴ (10 , 200) and (15 , 175) are two points on the line which represent

  the linear equation between the cars' numbers and the weeks

  numbers

∵ Point (x1 , y1) is (10 , 200) and point (x2 , y2) is (15 , 175)

∴ x1 = 10 , x2 = 15 and y1 = 200 and y2 = 175

- Use the rule of the slope above to find m

∴

- Substitute the value of x in the form of the linear equation above

∴ y = -5x + c

- To find c substitute x and y by one the coordinates of one of the

  two points

∵ x = 10 when y = 200

∴ 200 = -5(10) + c

∴ 200 = -50 + c

- Add 50 to both sides

∴ 250 = c

- Substitute the value of c by 250

∴ y = -5x + 250, where the number of cars seen each week is y and

  x is the number of the week

∵ f(x) = y

∴ f(x) = -5x + 250

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: