Math+Investigation

Investigation: 7x + 11y = 100

//To investigate if there is a pattern or a rule with the x and y integers.//

At first I searched for both positive integers for x and y:

11+89=100 22+78=100 33+67=100 55+45=100 66+34=100 77+23=100 88+12=100 99+1=100
 * 44+56=100**

I could only find one postive integer that satisfied the equation; x: 8 y: 4

After finding only two positive integers I continued my search with positive and negative integers for x and y:

-11+111=100 -22+122=100 -44+144=100 -55+155=100 -66+166=100 -77+177=100 -88+188=100 -99+199=100
 * -21+121=100**
 * -33+133=100**
 * -98+198=100**

I found 4 different integers that satisfied the equation:

1) 44+56=100 x=8 y=4

2)-21+121=100 x=-3 y=11

3)-33+133=100 x=19 y=-3

4)-98+198=100 x=-14 y=18

After observing the values of x and y I listed them in order:

Values:
 * x y**

-14 18 -3 11 8 4 19 -3 30 -10 41 -17 52 -24

After seeing the values placed in order I noticed a pattern between the values increasing as decreasing, in x, from 8 to 19 it added 11 and from 19 to 30 also. In y, from 4 to -3 it subtracted 7 as in -3 from -10 it subtracted 7. Therefore I could make a rule:

x = 8 + 11(n - 1) y = 4 - 7(n - 1)

Proving the rule:

n = 4

x = 8 + 11(4-1) y = 4 - 7(4-1)

x = 8 + 11(3) y = 4 - 7(3)

x = 8 + 33 y = 4 - 21

x = 41 y = -17

7(41) + 11(-17) = 100

287 + -187 = 100

After testing the rule I could see that it is right and there are infinite integers for 7x + 11y = 100