Math+Investigation

In this page I will describe the math investigation.

We were asked to write 100 as the sum of two integers, one divisible by 7 and the other divisible by 11. We had to use our answer to find formulas giving all the solutions of the following equation where x and y are integers.

First, I made up a list of all the multiples of 7 until 210, then next to them I put the number that I needed to add to get 100. E.g:

7 + 93 14+ 86 21+ 79 28+ 72 35+ 65

I knew that one number had to be a multiple of 7 and the other one of 11, and all the numbers in the first column were multiples of 7. So, I started to look for multiples of 11 on the second column and the first one I found was 44. The addition looked like this: 56 + 44 = 100. Next to the addition I put this: (7x8)+(11x4)=100. So, my first value for x was 8 and for y was 4. I looked for multiples of 11 until 210. I had this: x = 8, 19, 30. y = 4, -3 and -10.

After that, I did the same thing but instead of starting with a list of multiples of 7 it was a list of multiples of 11. The values I got were: x = 8, -3, -14, -25 y = 4, 11, 18, 25

They also asked us to try with negative numbers. Like instead of starting with 7, staring with -7. I did the same process I did before, here are the values: x = -3, -14, -25, -36 y = 11, 18, 25, 32

Later, I started the column with -11. Here are the results I got: x = 19, 30, 41, 52 y = -3, -10, -17, -24

After doing all this, I listed all my results in order, from smallest to biggest. The table looked like this: -25 -14 -3 8 19 30 41 52 || -24 -17 -10 -3 4 11 18 25 32 ||
 * X || Y ||
 * -36

After looking at the table I discovered a pattern. The difference between the values in column X was 11. If you go down you just have yo add 11 to get the next result, and if you go up you just subtract 11. And in column Y the difference between the values is 7. So, if you go down you just add 7 and if you go up you just subtract 7.

Also, let's say you subtract the first two values -36 - -24 = -12, and when you subtract the values below, the answer is -8, and if you subtract the next values the answer is -4. So you can see that the difference between them is 4. So, if you already have one value of x and one values of y, to get the next values use this formula:

1. if you want to find the next values, but that are are lower than the values you already have, you do this: (x - y)-4 2- if you want to find the next values, but that are bigger than the values you already have, you do this: (x - y)+4