Grades 11-12 Video Solutions 2011-11&12 Video Solutions 2011 problem17

11&12 Video Solutions 2011 problem17

Video Transcription

Video Summary

The problem involves finding the number of real number pairs \((x, y)\) that satisfy a given equation. By expanding both sides and utilizing a change of variable approach, it was determined that the equation simplifies into a conic form with a rotation. Using new variables \(u = x + 5\) and \(v = y - 5\), the equation simplifies to \(u^2 + uv + v^2 = 0\). By examining possible solutions, it is shown that both \(u\) and \(v\) must equal zero for real solutions, leading to exactly one pair of \((x, y)\) that satisfies the equation: \((0, 0)\). The answer is one solution.

Keywords

real number pairs

conic form

change of variable

equation simplification

unique solution