Let

x = the number of video games Jeff had

y = the number of video games Mike had

Jeff said to his younger brother, mike "give me 8 of your video games, and then I'll have twice as many games as you have."

x + 8 = 2(y - 8)

Mike said, "No way! You give me 8 of your games, and then we'll have the same number of games."

y + 8 = x - 8

We have the following system of equations:

x + 8 = 2(y - 8)

y + 8 = x - 8

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x = 56 games

y = 40 games

Jeff had 56 games and Mike had 40 games.