Agroup of ladies meet for an afternoon tea party. they bring all their cats. in all, there are 26 heads and 84 feet. how many ladies and how many cats are in the room

X= ladies       y = number of cats 

Each lady has 2 feet and each cat has 4 feet 

<=> 

x + y = 26 (1) => number of characters 
2x + 4y = 84 (2) => number of feet 

<=> operation : (1) * 2 

2x + 2y = 52 (1') 
2x + 4y = 88 (2') 

<=> operation : (2') - (1') 

2y = 26 

<=> 

y = 20 and x = 32- 16 = 16

10 ladies and 16 cats

H=p+c  (heads equals the sum of people and cats) since there are 26 heads:

p+c=26, solve for c, subtract p from both sides

c=26-p, now for feet...

2p+4c=f, using c found above gives you:

2p+4(26-p)=f and we are told that there are 84 feet so:

2p+4(26-p)=84  perform indicated multiplication on left side

2p+104-4p=84, combine like terms on left side

-2p+104=84  subtract 104 from both sides

-2p=-20  divide both sides by -2

p=10, and since c=26-p

c=26-10

c=16

So there are 10 people and 16 cats in the room.