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Algebra Difficulty 3.0 AMC 10/12 Prove it United Kingdom

Solve the pair of simultaneous equations
$x^2 - 2xy = 1,$
$5x^2 - 2xy + 2y^2 = 5.$

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.