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Heat engine X takes four times more energy by heat from the hot reservoir than heat engine Y. Engine X delivers two times more work, and it reject seven times more energy by heat to the cold reservoir than heat engine Y. Find the efficiency of heat engine X and heat engine Y.

Qin,x - Qout,x = Work,x
Qin,y - Qout,y = Work,y
Work,x = 2 Work,y
Qin,x = 4 Qin,y
Qout,x = 7 Qout,y
4Qin,y - 7Qout,y = 2 Work,y
7Qin,y - 7Qout,y = 7 Work,y
3Qin,y = 5Work,y
Since Efficiency = W/Qin,
Efficiency,y = 3/5

Work,x/Qin,x = 2Work,y/4Qin,y
Efficiency,x = (1/2)Efficiency,7 = 3/10

I got everything up until i got to the
7Qin,y - 7Qout,y= 7 Work,y
where did those numbers come from?

It is the same as the second equation on the list, with both sides multiplied by 7. It was one step in elimimating the Qout,y variable when combined with the equation above it.

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