Maths+Paper+6

Calixto Hernandez 9A

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

<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;">Negative X positive Y <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 <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;">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;">121 <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;">,132,143,154,165,176,187, <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;">198 <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 7 <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;">-7,-14, <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;">-21 <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;">,-28,-35,-42,-49,-56,-63,-70,-77,-84,-91, <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;">-98 <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=18 <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;">X=-14

<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=11 <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;">X=-3

<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;">Positive X Negative Y <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 7 <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;">105,112,119,126, <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;">133 <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;">,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 <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;">-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 <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;">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 <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;">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 <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;">X= -14,-3,8,19,30 X=X+11 <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;">Y=-10,-3,4,11, 18 Y=Y+7

<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;">Explanation: <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;">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.

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

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