The proof of equivalent formulas of ridge regression

Let's define $ \hat{x} $ as the optimal solution of the first problem and $ \tilde{x} $ as the optimal solution of the second problem. Namely you can always have a pair of $ t $ and $ \lambda \geq 0 $ such the solution of the problem is the same. How could we find a pair? Both problems are Convex and smooth so it should make things simpler. Pay attention that the 2 base equations are equivalent.