Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 21, 2011"
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==Solution== | ==Solution== | ||
− | + | Note: someone check my arithmetic please | |
+ | |||
+ | For purposes of generalization, let the equations be <math>ax+by=1002001</math> and <math>cx+dy=2004002</math>. Notice that <math>2*1002001 = 2004002</math>. Swap this out for a new variable <math>z</math>. | ||
+ | This gives <math>ax+by=z, cx+dy=2z</math>. | ||
+ | |||
+ | Multiply the left equation by two and substitute it into the other equation. | ||
+ | |||
+ | <math>2ax+2by=cx+dy</math>, which implies that<math>(c-2a)x=(2b-d)y</math>. Substituting the actual numbers back in gives: | ||
+ | <math>(997997-686686)x=(630630-8008)y</math> | ||
+ | Simplifying: | ||
+ | <math>311311x=622722y</math> | ||
+ | Which further simplifies to: | ||
+ | <math>x=2y</math> | ||
+ | |||
+ | Therefore: | ||
+ | |||
+ | Sample solution: <math>x=2, y=1</math> | ||
+ | |||
+ | General Solution: <math>x=2n, y=n</math> |
Revision as of 15:57, 21 July 2011
Problem
AoPSWiki:Problem of the Day/July 21, 2011
Solution
Note: someone check my arithmetic please
For purposes of generalization, let the equations be and . Notice that . Swap this out for a new variable . This gives .
Multiply the left equation by two and substitute it into the other equation.
, which implies that. Substituting the actual numbers back in gives: Simplifying: Which further simplifies to:
Therefore:
Sample solution:
General Solution: