Question

In one state, speeding fines are determined by `F = 6(x - 60)+100`, where `F` is the cost, in dollars, of the fine if a person is caught driving `x` miles per hour. If the fine comes to $178, how fast was the driver going?

Answer 1

`73" miles per hour"`

Explanation:

`"we require to solve the equation for x"`

`6(x-60)+100=178`

`rArr6x-360+100=178`

`rArr6x-260=178`

`"add 260 to both sides"`

`6xcancel(-260)cancel(+260)=178+260`

`rArr6x=438`

`"divide both sides by 6"`

`(cancel(6) x)/cancel(6)=438/6`

`rArrx=73`

Answer 2

If the fine comes to $178, the driver was going 73 mph

Explanation:

`F=6(x−60)+100`

This equation means:
For every mile over 60 mph
you are charged $6
plus a fixed surcharge of $100

`F...= . . 6 . . . . . . . .(x−60). . . . . . .+ . . . . . . . . .100`
Fine = . . $6 . . for each mph over 60 . . plus . . . . fixed surcharge

That's why it says `(x - 60)`.
You get the first 60 miles free, and you are fined only for the mph over that.

To determine how many miles per hour that fine should be for,
60 mph is subtracted from `x`, which is your clocked speed.

Then you are fined $6 for each mph over the first 60, and that's why it says `6(x - 60)`.

So if you are ticketed for driving 61 mph, your fine is
1 mph overage @ $6 ea `=` $6
plus the surcharge of $100
$106

If you are ticketed for driving 70 mph, your fine is
10 mph over 60   @ $    6 ea . . . $  60
1 fixed surcharge @ $100 ea. . . $100
............................................................................
Total fine . . . . . . . . . . . . . . . . . . . . .$160
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

You can solve these problems just by using the formula.
`F=6(x−60)+100`

Example:
What is the fine for a driver clocked at 81 mph?

Sub in 81 and solve for `x`
`F=6(81−60)+100`
.......................

You can also use the equation backwards.

If the fine is $178, how fast were you going?

1) There is a fixed surcharge of $100.
So the fine for "excess mph" is $78

2) That is $6 for every mile over 60
$78 `-:` $6 = 13 `larr`mph over 60
Speed = 60 mph + 13 mph more

Answer:
You were going at a speed of 73 mph
.....................

Sometimes it's easier just to sub in the given number

`F=6(x−60)+100`
Sub in $178 for Fine and solve for `x` (mph)

`178 = 6(x - 60) + 100`

1) Clear the parentheses
178 = 6`x` - 360 + 100

2) Combine like terms
178 = 6`x` - 260

3) Add 260 to both sides to isolate the 6`x` term
438 = 6`x`

Divide both sides by 6 to isolate `x`, already defined as the speed
73 = `x`

Answer:
You were going at a speed of 73 mph `larr` same answer
...........................

Check
13 over-60 miles @ $     6.00 ea . . . . .$  78
One surcharge     @ $100.00 ea . . . . .$100
........................................................................................
Total fine . . . . . . . . . . . . . . . . . . . . . . . . . .$178
Check!