Why Do Terms in the Energy Equation Not Cancel as Expected?

[2much]

I have an equation relating potential energy₁ plus gravitational energy₁ = potential energy₂ plus gravitational energy₂

1/2 mv²₁ + mg₁ = 1/2 mv²₂ + mg₂

Now cancelling out the terms I have

v²₁ + 2gy₁ + v²₂ 2gy₂

Now I don't understand why mg in the second term of both expressions don't cancel out. Why are we left with 2gy, not just y

 

[CompuChip]

First of all, note that gravitational energy is mgh, mg is just the force.
So you presumably meant

1/2 m v₁² + mgy₁= 1/2 m v₂² + mgy₂

Then you can multiply by 2 and divide it by m, but of course then you have to multiply everything, which gives

2/m(1/2 m v₁² + mgy₁) = 2/m (1/2 m v₁²) + 2/m (mgy₁) = v₁² + 2gy₁.
The 2 and 1/2 cancel out in the first term, and the m cancels out in both terms, but you have to leave the rest.
The right hand side is similar.