Melody

Using matrices

w=-3   x=-7    y=8     z=14

Just say the boys DO sit together

they can be in  pos 1&2 or 2&3 or 3&4 or 4&5 or 5&5 or 6&7 or 7&8    that is 7 ways.  

But either boy can be first so I have to times that by 2.

14 ways for the boys to sit together.   Once they are sitting there are 6! ways to sit the girls 

14*6! = 10080 ways for the boys to sit together.

With no restrictions there are  8!  ways to sit the kids,

With the restrictions, there are     8! - 16*6!  =  6!(56-16) = 6!*40 =   28 800 ways.

Well the second does not appear to have a solution

The first seems to be all points such that

 y = 2z-1 = 2x-1      (That is  z=x)

Try drawing it on a reasonably accurate graph.

1/2  for the first  *  1/2 for the second  = 1/4

ok, well I do not think the question makes sense. 

Here is what WolframAlpha has to say:

I started doing this by saying

\((x-a)(x-b)(x-c)(x-d)(x-e)=x^5+7x^4-2\)

a+b+c+d+e= -7    and       abcde=2    

ab     +ac  +ad  +ae  +bc  +bd  +be  +cd  +ce  +de    = 0

abc+abd+abe+acd+ace+ade+ bcd+bce+bde+cde   = 0

abcd + abce + abde +acde +  bcde =0

Maybe you can use that.  

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So next I used calculus to find the turning points of    \(y=x^5+7x^4-2\)

I found that there are only 2 turning points.  One at x=0 and the other at x=-5.6

Since there are only 2 points there can be a maximum of 3 distinct real roots.

 So the other 2 roots are not real but they are the conjugates of one another.   ie   p+qi   and   p-qi

If you let  a=p+qi   and  b=p-qi       then ab=p^2+q^2

I am just thinking in print.

Maybe you can get some ideas from this.

ok I have responded.

If you want to ask more questions then ask on the original AND leave a note here saying that you have done so.  :)

MGoo said on a repost that he does not understand this answer.

At any one time there are 4 tires being used.

Since there are 5 tires in total that means that each tyre must be used 4/5 of the time.   (4 are used out of a possible 5 tyres)

4/5 * 20000 = 16000

Each tyre does 16000 km.

Thanks abcd...  Now it looks relevant.

I changed the colours so it is easier to see both graphs.