We present short and elementary proofs of the Karush-Kuhn-Tucker Theorem for problems with nonlinear inequality constraints and linear equality constraints, the Lagrange Multiplier Theorem for equality-constrained optimization problems, and an Implicit Function Theorem. Most proofs in the literature rely on advanced concepts of analysis and optimization, whereas elementary proofs tend to be long and involved. By contrast, the presented proofs use only basic results from linear algebra and optimization. Our proof of the Karush-Kuhn-Tucker Theorem is based only on facts from linear algebra and the definition of differentiability. The proofs of the Lagrange Multiplier and Implicit Function theorems use a similar approach and are based on the critical-point condition for unconstrained minima (Fermat's Rule), and a single application of the Weierstrass theorem. The simplicity of the proofs makes them particularly suitable for use in a first undergraduate course in optimization or analysis, and also for courses not aimed specifically at optimization theory, such as mathematical modeling, mathematical methods for science or engineering, problems seminars, or any other circumstance where a quick treatment is required.