Math+(investigation)

Write 100 as the sum of two integers, one divisible by 7 and the other divisible by 11. Use your answer to find formulas ginvin all the solutions of the following equation where x and y are integers.
So 7x+11y=100

**Investigation paper 6(charts file)** Enter the file and you will find 3 charts each chart is done with the same technique but chart 1 is positve and positive numbers chart 2 negative and positive and at last chart 3 of positive and negative numbers. That was how I was able to find the number 100 and they are highlighted in red.



=What are the values of x and y?=

====In chart one i was only able to find one number 100 because those two integers are the only positive numbers that add up to 100 and if i continued with other integers they would add up to -100 and that was not what i was looking for.====

=Do you notice a pattern?=

Yes, as you can see the pattern in chart one is -11.
= = =**What are the values of x and y for chart 2?**=

The values of x and y for chart 2 are:

x=133, 210,287,364,441,518,595,672,∞ y=-33,-110,-187,-264,-341,-418,-495,-572, ∞

In chart two i was able to find many integers that add up to 100, as you can see in chart two the pair of integers are made by a positive integer and a negative integer not as chart one that the were formed by two positive integers. I could have continued with the multiplications but it was long enough for me to prove that there are many integers that add up to 100 and the chart continues up to infinite.

= Do you notice a pattern? =

yes in chart two i also notice a pattern the pattern is adding 7 each time from one integer to another.

= What are the values of x and y for chart 3? =

The values of x and y for chart 3 are:

x= -21, -98,-175,-252,-329,-406,-483,-560, ∞ y= 121,198,275,352,429,506,583,660, ∞

In chart three i was able to find many integers also that added up to 100, as you can see in chart three the pair of integers are made by a positive integer and a negative integer as it was in chart two but vice-versa. I could have continued with the multiplications but it was long enough for me to prove that there are many integers that add up to 100 and the chart continues up to infinite.

=Formula:=

In this step now i am writing my final formula that is made up of the pattern of integers that i found before.

Xn= 8-11(n-1)

Yn= 4+7(n-1)

testing my formula:

N=16

x=8-11(16-1) = 15 = -11x15 = -165 = 8-165 =-157 =-157x7 = -1099

y=4+7(16-1) =15 =15 x 7 = 105 = 105+4 =109 =109 x11 = 1199

1199-1099= 100

Now i have proved that this is the only formula that works.