Question 1205255
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x = number of small tables
4x = cost of just those small tables only
The coefficient 4 means "it costs $4 per small table".


y = number of large tables
6y = cost of just those large tables only
The coefficient 6 means "it costs $6 per large table".


4x+6y = total cost
4x+6y = 180 means you want the total cost to be $180


One solution, of many, is x = 0 and y = 30
x = 0 small tables costs 4x = 4*0 = 0 dollars
y = 30 large tables costs 6y = 6*30 = 180 dollars
total cost = 0+180 = 180


Another solution would be x = 3 and y = 28. 
Each time x goes up by 3, y drops by 2. This is from the slope -2/3. Note that solving 4x+6y = 180 leads to y = (-2/3)x+30.
If x = 3 small tables and y = 28 large tables are rented, then you spend a total of 4*x+6*y = 4*3+6*28 = 12+168 = 180 dollars.


Here is a list of all possible nonnegative integer solutions
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(0, 30), (3, 28), (6, 26), (9, 24), (12, 22), (15, 20), 
(18, 18), (21, 16), (24, 14), (27, 12), (30, 10), (33, 8), 
(36, 6), (39, 4), (42, 2), (45, 0)
</pre>There are 16 solutions in that list. Each solution is of the form (x,y).
The list was generated using the sequence command in GeoGebra.
All of the points mentioned are on the line 4x+6y = 180 aka 2x+3y = 90 aka y = (-2/3)x+30.
Desmos or GeoGebra or similar can be used to graph.
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