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( ax + by = c ) Solutions exist iff ( \gcd(a,b) ) divides ( c ). Here gcd(7,3)=1, divides 50 → solutions exist.
( x = x_0 + \frac{b}{\gcd(a,b)} \cdot t ) ( y = y_0 - \frac{a}{\gcd(a,b)} \cdot t ) for integer ( t ). Slide 8: Final Slide – Discussion Question If the jars held 6L and 4L, total 50L: ( 6x + 4y = 50 ) → divide 2: ( 3x + 2y = 25 ) gcd(6,4)=2 divides 50 → solutions exist.
(Answer: e.g., x=1, y=11; x=3, y=8; x=5, y=5; etc.) “Like Kaleb, you can solve real puzzles with Diophantine thinking.”
