Please complete the math and explain

Let x be the mass of the liquid itself, and y the mass of the empty bottle. Then

x + y = 1  1/2 kg

x = y + 3/4 kg

Substitute x into the first equation and solve for y :

(y + 3/4 kg) + y = 1  1/2 kg

(y + 3/4 kg) + y = 3/2 kg

2y + 3/4 kg = 3/2 kg

2y = 3/2 kg - 3/4 kg

2y = 6/4 kg - 3/4 kg

2y = (6 - 3)/4 kg

2y = 3/4 kg

y = 3/8 kg

Now solve for x :

x = 3/8 kg + 3/4 kg

x = 3/8 kg + 6/8 kg

x = (3 + 6)/8 kg

x = 9/8 kg

That is, the empty bottle weighs 3/8 kg, and the liquid that fills the bottle 3/4 of the way weighs 9/8 kg.

If the bottle is completely filled, then it contains an amount of liquid weighing z such that 3/4 of z weighs 9/8 kg. Solve for z :

3/4 z = 9/8 kg

z = (9/8)/(3/4) kg

z = (9/8)•(4/3) kg

z = (9•4)/(8•3) kg

z = 36/24 kg

z = 3/2 kg

So, the total weight of the completely filled bottle is

y + z = 3/8 kg + 3/2 kg

y + z = 3/8 kg + 12/8 kg

y + z = (3 + 12)/8 kg

y + z = 15/8 kg

c. (3,2,6) is the answer.