math

NRICH - "In Particular"

The problem: Write 100 as the sum of two integers, one divisible by 7 and the other divisible by 9. Use your answer to find formulas giving all the solutions of the following equation where x and y are integers.

So: 7x+11y=100 I only found one pair of positive integers which is: 56+44=100   To find it, I wrote multiples of 7 from 7 to 98, and multiples of 11 from 11 to 99; so I started looking at them, until I found the only pattern that there was in the positive integer. The values of x and y are: x=8 y= 4 Since 8x7=56, and 4x11=44. So 56+44=100 I found 3 negative integers which are: -21+121=100   -98+198=100    -175+275=100    To find them, I wrote the multiples of 7, from -7 to -182, and the multiples of 11, from 11 to 286; so I started looking at them, until I found these 3 patterns, which were in the negative integers. The values of x and y are: x=-3 y=11 -3x7=-21, and 11x11=121. So -21+121=100 x=-14 y=18 -14x7=-98, and 18x11=198. So -98+198=100 x=-25 y= 25 -25x7=-175, and 25x11=275. So -175+275=100 I noticed that there is a pattern, which is that in the x terms, the difference between all the following x terms, is -11, and the difference between all the following y terms is +7. n-11=x n+7=y Here is the equation that I made: // x= -11n+19 // // y //// =7n+3 // N is any number. So to find the pattern in x, you have to substitute n by any number, then multiply it by -11, the answer add it by 19. That’s how you get a value of x. Do this again with the number that follows the one that you substituted by n and when looking at the answers, you will find out that the difference between these y terms is -11, so the equation works with any number. To find the pattern in y, you have to substitute n by any number, and then multiply it by 7, the answer add it by 3. That’s how you get a value of y. Do this again with the number that follows the one that you substituted by n and when looking at two answers, you will find out that the difference between these y terms is +7, so the equation works with any number.
 * 1) First work with positive integers to see how many pairs of values you can find to satisfy the equation.
 * 1) What are the values of x and y?
 * 1) Now try negative integers for x with positive integers for y and vice-versa.
 * 1) List your values of x and y in an order.
 * 1) Do you notice a pattern?
 * 1) Can you make equations that describes how you get from one value of x and y to the next value?


 * 1) If you can, test to see if your equations work for several other values of x and y.

Here I tested if my equation matched with the pattern I noticed, or not:

This one is for x terms:

n=4 -11x4= -44 -44+19= -25   -25    -11    n=5 -11x5= -55 -55+19= -36   -36    -11    n=6 -11x6= -66 -66+19= -47   -47   So the difference between each x term is -11.

This one is for y terms:

n=4 7x4= 28 28+3= 31 31

+7

n=5 7x5= 35 35+3= 38 38

+7

n=6 7x6= 42 42+3= 45 45

My equation worked, the difference between each x term is -11, and the difference between the y terms is +7. To find out it they worked all I did was to substitute n, and then I followed the equation of x/y to find the value. I used three examples in x, and three examples in y. When I found the answers, I looked at the difference between each value, to see if my equation worked and it worked for both x and y.