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: 
