Final answer:
Liam has a total of 66 chairs. We set up two equations based on the fact that he has 3 chairs left over when arranged in 9 rows and 19 left over when arranged in 7 rows. We found a common solution that satisfies both equations.
Step-by-step explanation:
To determine how many chairs Liam has, we need to set up equations based on the information given. Firstly, if he sets up the chairs in 9 rows with some number of chairs per row, he has 3 chairs left over. This can be written as n = 9x + 3, where n is the total number of chairs and x is the number of chairs per row.
Secondly, if he sets up the chairs in 7 rows with the same number of chairs per row, he has 19 left over. This gives us a second equation: n = 7x + 19.
To find the total number of chairs n, we need a value of x that makes both equations work. We are looking for the smallest number that, when multiplied by 9, has a remainder of 3, and when multiplied by 7, has a remainder of 19. By checking multiples of 9 and adding 3 (finding numbers of the form 9k+3 where k is a whole number), we can see if there's a corresponding multiple of 7 that is 19 less than one of those numbers. The smallest such number that satisfies both conditions is 66. So, Liam's total number of chairs is 66.