Can someone please help me with this....can't figure out how to set it up :)

Question

texaschic101 · Answer

In the Newham hockey league a team scores 5 points for a win, 2 for a draw and 0 if they lose. the club has played 20 games in the league and has scored 21 points. what is the smallest possible number of games the team could have lost

anonymous · Answer

woa im a little confused on this problem......one sec :/

texaschic101 · Answer

I started with this..
5w + 2d + 0l = 21
but then I do not know what to do....

texaschic101 · Answer

w + d + l = 20 ??

texaschic101 · Answer

should I just start subbing in numbers for l, starting with 1 and going up....trial and error ?

ganeshie8 · Answer

I think smallest number of losses is 11

texaschic101 · Answer

11 loses. leaving 9 games.
1 win = 5 points
8 draws = 16 points
20 games at 21 points
that works

texaschic101 · Answer

is there any other way to do this except for trial and error ?

ganeshie8 · Answer

5w + 2d = 21

w + d + l = 20

ganeshie8 · Answer

from the first equation, 
solve $d$ and substitute it in second equation

texaschic101 · Answer

d = -5/2w + 21/2

w + (-5/2w + 21/2) + l = 20
w - 5/2w + 21/2 + l = 20
2/2w - 5/2w  = 20 - 21/2 + l
wait...where do I put the l

ganeshie8 · Answer

get rid of fractions first, they look clumsy

texaschic101 · Answer

2/2w - 5/2w = 20 - 21/2 + l --->(2)
2w - 5w = 40 - 21 + l

ganeshie8 · Answer

w + d + l = 20


2w + 2d + 2l = 40

2w + 21 - 5w + 2l = 40

-3w + 2l = 19

2l = 19 + 3w

ganeshie8 · Answer

we want minimum value for Losses,

so put w=0, that gives u minimum value for Losses.

but can u really put w=0 ?

ganeshie8 · Answer

Nope, we cannot put w = 0, cuz, that will give fraction for Losses.

so, put w = 1

texaschic101 · Answer

21 - 19 = 3w
3 = 3w
1 = w

ganeshie8 · Answer

2l = 19 + 3*1

2l = 22

l = 11

ganeshie8 · Answer

its 2L, not 21 :|

texaschic101 · Answer

oh....I see now. I was trying to use elimination...couldn't get it to work. Your a genius

ganeshie8 · Answer

$w + d + l = 20$

$2w + 2d + 2l = 40$

$2w + 21 - 5w + 2l = 40$

$-3w + 2l = 19$

$2l = 19 + 3w$

ganeshie8 · Answer

now see, the $l$ and $1$ look same when we dont use latex lol

ganeshie8 · Answer

no, you're a genius :)

texaschic101 · Answer

I see that....however, I am not too good with Latex....lol

texaschic101 · Answer

thank you so much for your help :)

ganeshie8 · Answer

np :)