Answer:
5
Explanation:
You have more than 1 each of 3- and 4-legged tables that have a total of 23 legs, and you want to know the number of 3-legged tables.
Equation
Let t and f represent the numbers of three- and four-legged tables respectively. Then we require ...
3t +4f = 23 . . . . and t > 1, f > 1
Solution
The value 23 is clearly 5·4 +1·3, so we have one solution (t, f) = (1, 5). Increasing the number of 3-legged tables by 4 and decreasing the number of 4-legged tables by 3 will leave the number of legs unchanged. Then for some 'n', we have ...
t = 1 +4n
f = 5 -3n
The only solution that has t > 1 and f > 1 is the one for n=1:
t = 1 +4·1 = 5 . . . . a total of 15 legs
f = 5 -3·1 = 2 . . . . a total of 8 legs, making 23 when added to 15.
There are 5 three-legged tables.
__
Additional comment
There are formal methods for finding integer solutions to equations of the kind we wrote here. For this problem, it isn't necessary to go to that trouble, as we can "guess" a solution quickly.
A graphing calculator can also be helpful for finding solutions to the equation. This is shown in the attachment. The black points mark the "floor" values of the variables of interest (x=t, y=f).