A diner has 19 booths. Some booths seat 4 people and some booths seat 5 people. Each booth in the diner is full and there are 84 people in the diner. How many booths are for 4 people?
Let x be the number of booths for 4 people and y be the number of booths for 5 people.
We know that:
x + y = 19 (since there are a total of 19 booths)
4x + 5y = 84 (since there are a total of 84 people and each booth is full)
We can solve for x in the first equation by subtracting y from both sides:
x = 19 - y
Substituting this into the second equation:
4(19-y) + 5y = 84
76 - 4y + 5y = 84
y = 8
Substituting this value of y back into x = 19 - y:
x = 19 - 8 = 11
Therefore, there are 11 booths for 4 people.