This is a logic problem about tennis.

[Terrell]

Yes, you got it right, but having reached this point for case 2

you can solve for those two variables and discover an impossibility.

because $P_r$ cannot be 7/2 ? Also, in retrospect, I never thought that I need to set up an equation to work on this problem so are there problems that does not need setting up equations? thank you!

 

[haruspex]

because $P_r$ cannot be 7/2 ? Also, in retrospect, I never thought that I need to set up an equation to work on this problem so are there problems that does not need setting up equations? thank you!

Equations are just a convenient way of expressing reasoning. The Ancient Greeks managed without. Whether it is worth encoding a problem as equations depends on how succinct the notation is and how complex the problem.

 

[rcgldr]

Also, in retrospect, I never thought that I need to set up an equation to work on this problem so are there problems that does not need setting up equations? thank you!

This can be done logically. If Pat wins on Stacy's serve and Stacy win's on Pat serve, then the set's game score is the same as if Pat won on her serve and Stacy won on her serve. The key factor is how many times Pat won on Stacy's serve minus how many times Stacy won on Pat's serve. Assume game score is even and equal to the number of times Stacy won at some point in the set. Consider the case where the game score is Pat 4, Stacy 4, and that Pat wins the last two games, the set ends up Pat 6, Stacy 4 regardless of who served first. Now consider the case where the game score is Pat 3, Stacy 3, and that Pat wins the last three games. If Stacy served first, then Pat won both times Stacy served in the last 3 games with Pat winning on Stacy's serve 2 more times than Stacy won on Pat's serve, but 5 can't be split up so that the difference equals 2. If Pat served first, then Pat only won once on Stacy's serve in the last 3 games, Pat winning on Stacy's serve 1 more time than Stacy won on Pat's serve, and 5 can be split up so the difference equals 1, specifically 3 - 2 = 1.

In tennis when a player wins when the other serves it's called a "break".
A list of possible outcomes for a set won when the winner wins their 6th game:

6 4 => +1 break, doesn't mater who served first
6 3 => +1 break if winner served first, +2 breaks if loser served first
6 2 => +2 breaks, doesn't matter who served first
6 1 => +2 breaks if winner served first, +3 breaks if loser served first
6 0 => +3 breaks, doesn't matter who served first