How many different ways can you make change with a $3?

Perfume A (a) costs $11 per ounce and perfume B (b) costs $19 per ounce. So, the expressions, 11a and 19b represent the cost of each perfume in the 5-ounce bottle. Because the perfume maker wants the bottle to cost $71, add the expressions 11a and 19b and set the sum equal to 71:

11a + 19b = 71

You would first make a table to help sort out the info. The table will have 3 rows and those rows will be PerfA, PerfB, and mix. The table will have 3 columns and those will be, from left to right, # oz, cost, total (which is the product of the first 2 columns). In the first row first column we will put an x, since we don't know how many ounces of perfume A we have. The cost of perfume A is 11, so that number goes in the first row second column. The total for the first row is x * 11 which is 11x. Now we move to row 2. We put a y in the second row first column since we don't know how much of perfume B we have. We do know that it costs 19 per ounce, so 19 goes in row 2 column 2. The product of those 2 is 19y. In row 3 first column we put a 5 since the maker wants the mix of A and B to be 5 ounces. In row 3 column 2 we put 71, since the maker wants the mix to cost 71. The product of those 2 numbers is 355. If you go straight down the first column, you have that the number of ounces of A (x) + the number of ounces of B (y) = 5, or x + y = 5. That equation relates the number of ounces to each other that is in the mix. We are looking for the money equation, which is going straight down the 3 column. This column relates the cost of A and B to each other that is in the mix. Standard form for an equation is Ax + By = C, so for us that will be 11x + 19y = 355. There you go!