MATH

__NRICH “In Particular” ____Made by: Roberto Zelaya __ __The Problem: __ Write 100 as the sum of two positive 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 = 100**
 * **X ** ||  **Y **  ||
 * 7  ||  93   ||
 * 14  ||  86   ||
 * 21  ||  79   ||
 * 28  ||  72   ||
 * 35  ||  65   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">42  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">58   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">49  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">51   ||
 * __<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">56 __ ||  __<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">44 __  ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">63  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">37   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">70  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">30   ||


 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">B. I just found one positive integer. **

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">56/7 = 8 & 44/11=4


 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">2. **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">The values of **X** & **Y: -X**=8**-Y**=4


 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">3. **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;"> Now try negative integers for x with positive integers for y and vice versa.

|| || || || || || || ||
 * **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">X ** || **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">Y ** ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-7  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">107   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-14  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">114   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">- 21  || <span style="color: #f6cbcb; font-family: 'Times New Roman','serif'; font-size: 16px;">121  ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-28  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">128   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-35  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">135   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-42  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">142   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-49  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">149   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-56  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">156
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-63  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">163
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-70  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">170
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-77  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">177   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-84  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">184
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-91  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">191
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">- 98   ||   __<span style="font-family: 'Times New Roman','serif'; font-size: 16px;">198 __
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-105  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">205
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-112  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">212


 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">3.B. **

|| || || || || || || ||
 * **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">X ** ||  **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">Y **  ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">7  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-107   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">14  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-114   ||
 * <span style="color: #fcc5c5; font-family: 'Times New Roman','serif'; font-size: 16px;">21  ||  <span style="color: #fcc5c5; font-family: 'Times New Roman','serif'; font-size: 16px;">-121   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">28  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-128   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">35  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-135   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">42  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-142   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">49  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-149   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">56  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-156
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">63  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-163
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">70  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-170
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">77  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-177   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">84  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-184
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">91  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-191
 * <span style="color: #f9c8c8; font-family: 'Times New Roman','serif'; font-size: 16px;">98  ||  <span style="color: #f9c8c8; font-family: 'Times New Roman','serif'; font-size: 16px;">-198
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">105  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-205
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">112  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-212


 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">4. **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">List your values of x and y in an order.


 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">(Explanation)  ||  **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">X **  ||  **<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">Y **  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">(Explanation)   ||
 * || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">19   ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-3   || ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-11 à  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">8   ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">4   ||   ß <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">+7   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-11 à  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-3   ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">11   ||   ß <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">+7   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-11 à  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-14   ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">18   ||   ß <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">+7   ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-11 à  ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">-25   ||  <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">25   ||   ß <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">+7   ||

5. notice a pattern??

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">Yes there is a pattern, the difference between x values to find them is 11.

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">6. FORMULA

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">X=8+11(N-1)

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px;">Y=4-7(N-1)

TESTING THE FORMULA

8+11(2-1)

4-7(2-1)

X= 19

Y= -3

7(19) +11(-3) =100

133+-33=100

I test this formula many times to see that is correct!