Non-linear constrained problem transformation to equivalent un-constrained problem

Question

I have the following non-linear optimization problem:

min $f(x, y, z) = x + y + z$
s.t.
$x^2 + y = 3$
$x + 3y + 2z = 7$

Is there a way to transform this problem to an equivalent minimisation problem without any constraints?
I have a hunch that it has something to do with Lagrange multipliers but I really can't figure out how to start this one.

Thanks,
Louis

Answer 1

Guide:

We have $y=3-x^2$, that is whenever we see $y$, we can replace it by $3-x^2$.

Also, we can express $z$ as a function of $x$ and $y$.

Hence yes, we just have to perform a substitution to replace everything in terms of $x$.

The unconstrained problem is a quadratic minimization problem.