Let x = number of games played
x - 15 = 22
x - 15 = 22
Kathy currently has -15 marbles, and she needs to get up to 22 marbles. Let x equal the number of games the girls still have to play.
x+(-15)=22
x=37
This is not the final answer, though. This is only the number of times Kathy won. Remember that Jane has already won 15 games. You now have to find the total amount of games they played by adding each girl's number of wins together.
37+15=52 games total
Since Jane won 15 times, she received 15 marbles from Kathy. Let's denote the number of marbles Jane had at the start of the game as J. Therefore, at the end of the game, Jane would have J + 15 marbles.
Now, Kathy had 22 more marbles than when she started. Let's denote the number of marbles Kathy had at the start of the game as K. Therefore, at the end of the game, Kathy would have K - 15 marbles.
Given that Kathy had 22 more marbles than her starting amount, we can write the equation K - 15 = K + 22.
Now, let's solve the equation:
K - 15 = K + 22
Subtract K from both sides:
-15 = 22
As you can see, -15 is not equal to 22. Therefore, there is no solution to this equation.
Hence, there seems to be an error in the given information or the relationship between the number of games and the number of marbles won/lost is not accurately defined.