Look at the table. Make a conjecture about the sum of the first 15 positive even numbers.
1 answer:
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Check out the image attachment for the filled out table. There may be more than one way to fill out the table, but I did it in the way I learned in the past.
Let's work through the table one row at a time
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So we'll start with row 1
Row 1, column1: The value that goes here is 12 as each adult pays $12
Row 1, column2: You'll write 'a' without quotes here as there are 'a' adults ('a' is just a placeholder for a number)
Row 1, column3: Write 12*a or 12a here. Simply multiply the cost per adult ($12) with the number of adults (a).
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Now onto row 2
Row 2, column1: It costs $6 per young adult, so we write 6 here
Row 2, column2: There are y young adults. Write 'y' here without quotes
Row 2, column3: Write 6y here. Multiply the number of young adults with the price per young adult
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Now onto row 3
Now we add up the values per each column to get the column totals
Row 3, column1: The individual costs 12 and 6 add to 18. We won't use this value but it doesn't hurt to write it in. If it is confusing to add in, then just ignore this cell. The reason why we won't use this is because the number of adults (a) and young adults (y) is not necessarily the same. If we were guaranteed they were the same, then we could use this value. But again there's no guarantee. It's probably best to steer clear of this cell.
Row 3, column2: We have 'a' adults and 'y' young adults. So a+y people total. This total is 8 as we know a family of 8 had been registered. So we write 8 in this box as well. The two expressions a+y and 8 are equal to each other allowing us to form the first equation a+y = 8
Row 3, column3: The cost for all the adults is 12a dollars. Similarly it costs 6y dollars for just the young adults. Adding up the two subtotals we get 12a+6y as the total cost for everyone. We're told that the family paid a total of $66. So like with the previous box, we can equate the two expressions getting us the second equation to be 12a+6y = 66
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Again everything is summarized in the image attachment.
The two equations we pull away from that table are
a+y = 8
12a+6y = 66
which is the system of equations to set up
Let's work through the table one row at a time
------------------------------------------------------------------------------
So we'll start with row 1
Row 1, column1: The value that goes here is 12 as each adult pays $12
Row 1, column2: You'll write 'a' without quotes here as there are 'a' adults ('a' is just a placeholder for a number)
Row 1, column3: Write 12*a or 12a here. Simply multiply the cost per adult ($12) with the number of adults (a).
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Now onto row 2
Row 2, column1: It costs $6 per young adult, so we write 6 here
Row 2, column2: There are y young adults. Write 'y' here without quotes
Row 2, column3: Write 6y here. Multiply the number of young adults with the price per young adult
------------------------------------------------------------------------------
Now onto row 3
Now we add up the values per each column to get the column totals
Row 3, column1: The individual costs 12 and 6 add to 18. We won't use this value but it doesn't hurt to write it in. If it is confusing to add in, then just ignore this cell. The reason why we won't use this is because the number of adults (a) and young adults (y) is not necessarily the same. If we were guaranteed they were the same, then we could use this value. But again there's no guarantee. It's probably best to steer clear of this cell.
Row 3, column2: We have 'a' adults and 'y' young adults. So a+y people total. This total is 8 as we know a family of 8 had been registered. So we write 8 in this box as well. The two expressions a+y and 8 are equal to each other allowing us to form the first equation a+y = 8
Row 3, column3: The cost for all the adults is 12a dollars. Similarly it costs 6y dollars for just the young adults. Adding up the two subtotals we get 12a+6y as the total cost for everyone. We're told that the family paid a total of $66. So like with the previous box, we can equate the two expressions getting us the second equation to be 12a+6y = 66
------------------------------------------------------------------------------
Again everything is summarized in the image attachment.
The two equations we pull away from that table are
a+y = 8
12a+6y = 66
which is the system of equations to set up