Thread: 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(x₁)
s.t.
2*sqrt(x₂y)=64
y=2*sqrt(2x₃)
x₁+x₂+x₃=112

the professor reformulated the problem without explanation to
max_{x1, x2, x3} 4x₁
s.t.
16x₂²y₂²=64⁴
y²=8x₃
x₁+x₂+x₃=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

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