Answer:
Proofs are in the explantion.
Step-by-step explanation:
We are given the following:
1) 
for integer 
.
1) 
for integer 
.
a)
Proof:
We want to show 
.
So we have the two equations:
a-b=kn and c-d=mn and we want to show for some integer r that we have
(a+c)-(b+d)=rn. If we do that we would have shown that .
kn+mn = (a-b)+(c-d)
(k+m)n = a-b+ c-d
(k+m)n = (a+c)+(-b-d)
(k+m)n = (a+c)-(b+d)
k+m is is just an integer
So we found integer r such that (a+c)-(b+d)=rn.
Therefore, .
//
b) Proof:
We want to show 
.
So we have the two equations:
a-b=kn and c-d=mn and we want to show for some integer r that we have
(ac)-(bd)=tn. If we do that we would have shown that .
If a-b=kn, then a=b+kn.
If c-d=mn, then c=d+mn.
ac-bd = (b+kn)(d+mn)-bd
= bd+bmn+dkn+kmn^2-bd
= bmn+dkn+kmn^2
= n(bm+dk+kmn)
So the integer t such that (ac)-(bd)=tn is bm+dk+kmn.
Therefore, .
//
Answer:
They won 17 games and drew 4 games
Step-by-step explanation:
The given parameters are;
The number of points the major league soccer team finished with = 55 points
The number of games the soccer team played = 28 games
The number of losses the soccer team had = 7 losses
The number of points awarded for each win = 3 points
The number of points awarded for each tie = 1 points
The number of points awarded for each loss = 0 points
Let x represent the number of wins, y represent the number of draws, and let z represent the number losses
Therefore;
z = 7
x + y + z = 28
3·x + y + 7×0 = 55
Therefore, we have the following system of equations;
x + y = 21...(1)
3·x + y = 55...(2)
Which gives;
The inverse of the matrix is given as follows;
Therefore;
x = 17, y = 4
Answer:
9.3
Step-by-step explanation:
186/20 = 18.6/2 = 9.3
2 5/8 = 2 15/24
1 1/3 = 1 8/24
cups of condensed soup: (2 15/24) - (1 8/24) = 1 7/24 cups
