Math

Hello This Is My Math Course Work Of Paper 6 and the rules are simple for this, this are the rules i have to follow:
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. =So 7x + 11y equals 100=

In the positive integers the only numbers i could find to satisfy the equations where both numbers where positive are 8 & 4

when 8 x 7 = 56 and 11 x 4 = 44 giving me a total of 100.
I have then a list of multiples of 7 which are positive and gave me 100 when i added them with another integer, until i found 3 posible solutions to the equation where x is positive and y is negative and this is what i found:

7+93=100 14+86=100 21+79=100 28+72=100 35+65=100 42+58=100 49+51=100 56+44=100 63+37=100 70+30=100 77+23=100 84+16=100 91+9=100 98+2=100 105+-5=100 112+-12=100 119+-19=100 126+-26=100 133+-33=100 140+-40=100 147+-47=100 154+-54=100 161+-61=100 168+-68=100 175+-75=100 182+-82=100 189+-89=100 196+-96=100 203+-103=100 210+-110=100 217+-117=100 224+-124=100 231+-131=100 238+-138=100 245+-145=100 252+-152=100 259+-159=100 266+-166=100 273+-173=100 280+-180=100 287+-187=100

I have now made a list of numbers where multiples of 11 plus an integer will give me 100, until I found 3 possible solutions to the equation where y is positive and x is negative.

11+89=100 22+78=100 33+67=100 44+56=100 55+45=100 66+34=100 77+23=100 88+12=100 99+1=100 110+-10=100 121+-21=100 132+-32=100 143+-43=100 154+-54=100 165+-65=100 176+-76=100 187+-87=100 198+-98=100 209+-109=100 220+-120=100 231+-131=100 242+-142=100 253+-153=100 264+-164=100 175+-175=100

Now i have put the X and Y integers in order in a box to make it more simple to find a rule for it, this is where i found that for X every time you subtract 11 and for it to be 8 which is the number im basing myself on. I also notice for Y that every time you go up 7 and for it to go to my number where im basing myself on i have to subtract 3.


 * X ||= Y ||
 * 41 ||= -17 ||
 * 30 ||= -10 ||
 * 19 ||= -3 ||
 * 8 ||= 4 ||
 * -3 ||= 11 ||
 * -14 ||= 18 ||
 * -25 ||= 25 ||

I have found out 2 rules that satisfy the equation, one is for X and the other for Y. this are the rules:

Y= 7n-3 X= 11n+19

So now i have to prove that my rule works and for that i choose a random number and decided to try it out.

Try out 1. # -3
Y= 7(-3)-3= -24 X= -11(-3)+19= 52 Y= 7(52) X= 11(-24) 364+-264=100

so it works for this integer and i will do the same for 2 more numbers and test it out fully.

Try out 2. # -32
Y= 7(-32)-3= -227 X= -11(-32)+19= 371 Y= 7(371) X= 11(-227) 2597+2497= 100

Try out 3. #45
Y= 7(45)-3= 312 X= -11(45)+19= -476 Y= 7(-476) X=11(312) 3432+-3332= 100