Optimization: transformation of target function and conditions

In a recent microeconomics lecture I was confronted with the following problem:

max_{x1, x2, x3} 2*sqrt(x_{1})

s.t.

2*sqrt(x_{2}y)=64

y=2*sqrt(2x_{3})

x_{1}+x_{2}+x_{3}=112

the professor reformulated the problem without explanation to

max_{x1, x2, x3} 4x_{1}

s.t.

16x_{2}^{2}y_{2}^{2}=64^{4 }y^{2}=8x_{3}

x_{1}+x_{2}+x_{3}=112

and then solved by plugging the conditions into the target function. How can such a transformation be determined without changing the problem at hand? what is the logic or the reason behind this?

thanks for any clue

Re: Optimization: transformation of target function and conditions

If $\displaystyle 2\sqrt{x_1}$ is at a maximum, then so is $\displaystyle 4x_1=(2\sqrt{x_1})^2$. The changes made to the conditions are similar - taking the 4th power of both sides and squaring both sides.

- Hollywood