This System Looks Simple…But Many Students Miss the Trick | MOA Lesson 16

Описание: Hello math fans!

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In Lesson 16, advanced high-school students tackle a nonlinear system of reciprocal equations - a problem that looks intimidating at first, but can be resolved systematically using a substitution approach, algebraic manipulation, and logical reasoning.

The lesson objective is to find all real numbers x and y that satisfy:

one over x plus one over y equals five,
and one over x squared plus one over y squared equals seventeen.

Your challenge as students is clear:

👉 Can you determine all real solutions (x, y) that satisfy this system?

Instead of solving the problem using the two variables x and y, which would yield a tedious and long calculation, we adopted the substitution approach (or method).

In this lesson, we guide students through a clear and systematic method:

🟢 Apply a change of variables to simplify the nonlinear system
🟢 Rewrite the system in terms of u and v
🟢 Use algebraic identities to calculate u minus v and factor expressions
🟢 Solve the resulting linear systems step by step
🟢 Translate solutions back to the original variables x and y
🟢 Verify all solutions numerically to confirm correctness

This lesson is suitable for students aiming to strengthen:

Solving nonlinear and reciprocal systems of equations
Applying substitutions and algebraic identities effectively
Logical reasoning and step-by-step problem-solving
Analytical verification of solutions
Preparation for high-level math competitions including AMC, AIME, JEE Advanced, and national Olympiads
Structured approaches used in elite mathematical training worldwide

By the end of this video, advanced high-school students will:

Confidently solve nonlinear reciprocal systems of equations
Apply change of variables and algebraic identities strategically
Verify solutions both algebraically and numerically
Enhance logical reasoning, analytical thinking, and problem-solving skills
Strengthen structured reasoning and advanced algebraic problem-solving abilities.

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🟢 Advanced algebra and nonlinear systems
🟢 University-style and international math challenges
🟢 Step-by-step analytical reasoning
🟢 Techniques essential for competitive mathematics

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The Math Olympiad Academy Team

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