Math


 * Investigation Project Paper 6**

Write 100 as the sum of two integers, one divisible by 7 and the other divisible by 11. Use your answer to find formulas giving all the solutions of the following equation where X and Y are integers. What are the values of X and Y ?

So 7 X +11 Y =100


 * **Multiples 7** || **Multiples 11** ||
 * 7 || 11 ||
 * 14 || 22 ||
 * 21 || 33 ||
 * 28 || 44 ||
 * 35 || 55 ||
 * 42 || 66 ||
 * 49 || 77 ||
 * 56 || 88 ||
 * 63 || 99 ||
 * 70 || 110 ||

The integers that are colour Pink are the integers that satisfy the equation given.


 * **X (7)** || **Y (11)** ||
 * 1 || 8.45 ||
 * 2 || 7.81 ||
 * 3 || 7.18 ||
 * 4 || 6.54 ||
 * 5 || 5.90 ||
 * 6 || 5.27 ||
 * 7 || 4.63 ||
 * 8 || 4 ||
 * 9 || 3.36 ||
 * 10 || 2.72 ||
 * 11 || 2.09 ||
 * 12 || 1.45 ||
 * 13 || 0.81 ||
 * 14 || 0.18 ||
 * 15 || -0.45 ||
 * 16 || -1.09 ||
 * 17 || -1.72 ||
 * 18 || -2.36 ||
 * 19 || -3 ||
 * 20 || -3.63 ||
 * 21 || -4.27 ||
 * 22 || -4.90 ||
 * 23 || -5.54 ||
 * 24 || -6.18 ||
 * 25 || -6.81 ||
 * 26 || -7.45 ||
 * 27 || -8.09 ||
 * 28 || -8.72 ||
 * 29 || -9.36 ||
 * 30 || -10 ||

Now try negative integers for X with positive integers for Y and vice-versa.

The integers that are colour Pink are the integers that satisfy the equation given.


 * **Y (11)** || **X (7)** ||
 * 1 || 12.7 ||
 * 2 || 11.14 ||
 * 3 || 9.57 ||
 * 4 || 8 ||
 * 5 || 6.42 ||
 * 6 || 4.85 ||
 * 7 || 3.28 ||
 * 8 || 1.71 ||
 * 9 || 0.14 ||
 * 10 || -1.42 ||
 * 11 || -3 ||
 * 12 || -4.57 ||
 * 13 || -6.14 ||
 * 14 || -7.71 ||
 * 15 || -9.28 ||
 * 16 || -10.85 ||
 * 17 || -12.42 ||
 * 18 || -14 ||
 * 19 || -15.57 ||
 * 20 || -17.14 ||
 * 21 || -18.71 ||
 * 22 || -20.28 ||
 * 23 || -21.85 ||
 * 24 || -23.42 ||
 * 25 || -25 ||
 * 26 || -26.57 ||
 * 27 || 28.14 ||
 * 28 || -29.71 ||
 * 29 || -31.28 ||
 * 30 || -32.85 ||
 * 31 || -34.42 ||
 * 32 || -36 ||

**Do you notice a pattern?**

Yes, it took me a while to realize it but later I found out that in the first question the pattern is of -11 and in the second question the pattern is of +7.


 * Can you make equations that describes how you get from one value of** **X** **and** **Y** **to the next value?**

There are some various equations to explain how I got the values of X and Y but the only equation that works using just positive numbers is:

X =8 Y =4 <--- This means that you can only get one pair of positive integers to work with the equation 7 X +11 Y =100.

So 7x8=56 and 4x11=44 (56+44=100)


 * Why is this the only one that works using just positive integers?**

This equation ( X =8 Y =4) is the only one that works properly to find out just positive integers that add up to our equation 7 X +11 Y =100 because the other pairs of integers that could make it work properly are only one positive and one negative.

Examples: X = -14 Y = -14 So -14x7= -98 18x11= 198 198-98= 100

X =-25 Y = 25 So -25x7= -175 25x11= 198 275-175= 100

**Equation**

X n= 8-11(n-1)

After I thoroughly analized the question and checked the pattern for the X integer it had a patter of -11 between the next integers. I found out that the equation above this by later substracting -1,which then gave me a number that then I substracted by the number of the pattern -11. At the end that number I substracted to -8.

Y n= 4+7(n-1)

After I thoroughly analized the question and checked the pattern for the Y integer it had a pattern of +7 between the next integers. So then I subtracted -1, which gave me a number that then I added by the number of the pattern that was +7. At the end that number I added to positive 4.

**Proving that my equation works:**

N=6 X = 8-11(6-1) So 6-1=5 5x-11= -55 8-55= -47 -47x7= -329 Y = 4+7(6-1) So 6-1=5 5x7=35 35+4=39 39x11=429 Y+X = 429-329= 100