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Paper 6 Investigation

The Problem 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 where x and y are integers. So 7x+11y=100

Positive integers Multiples of 7, 7,14,21,28,35,42,49, 56 ,63,70,77,84,91,98 Multiples of 11 11,22,33, 44 ,55,66,77,88,99 Y=4 X=8

Negative X positive Y Multiples of 11 110, 121 ,132,143,154,165,176,187, 198 Multiples of 7 -7,-14, -21 ,-28,-35,-42,-49,-56,-63,-70,-77,-84,-91, -98 Y=18 X=-14

Y=11 X=-3

Positive X Negative Y Multiples of 7 105,112,119,126, 133 ,140,147,154,161,168,175,182,189,196,203, <span style="background-color: transparent; color: #0000ff; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">210 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">Multiples of 11 -11,-22, <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">-33 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">,-44,-55,-66,-77,-88,-99, <span style="background-color: transparent; color: #0000ff; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">-110 <span style="background-color: transparent; color: #ff0000; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">Y=-3 X=19

<span style="background-color: transparent; color: #0000ff; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">Y=-10 X=30

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline;">All values in order Equations X= -14,-3,8,19,30 X=X+11 Y=-10,-3,4,11, 18 Y=Y+7

Explanation: Beginning with the positive integers is the best thing to, you get an idea of which multiples are the best. But when I combined negative integers and positive integers I got more results which gave me more ideas of a possible equation. At the beginning of the solving of the equation I tried different formulas which had worked for me in previous investigation like 2x+11, 2y+6 but none of my multiplications seemed to work. I tried squaring the number and subtracting but like the previous equations nothing worked. Finally I decided to add the number that gave me the result of the second integer which seemed to work for all the numbers in X pattern. I tried it with the Y pattern and it worked perfectly. On the next step I will use my sequence to predict the following integers which would add up to 100.

X=41 -187 X=-25 -175 X=52 364 X=-36 -252 Y=-17 +287=100 Y=25 +275=100 Y=-24 -264=100 Y= 32 352=100

As you can see by these results my two equations are correct and can be used for predicting further results of this problem