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 giving all the solutions of the following equation where x and y are integers. So 7x+11y=100

1. First work with positive integers to see how many pairs of values you can find to satisfy the equation

7x + 11y = 100 56 + 44 =100 2. What are the values of x and y? x = 8 y = 4

3. Now try negative integers for x with positive integers for y and vice-versa. 133-33=100 x=19 y= -3 210-110=100 --- x=30 y= -10 287-187=100 --- x=41 y=- -17 364-264=100 --- x=52 y= -24 -21+121=100 --- x= -3 y= 11 -175+275=100-- x= -25 y= 25 -252+352=100-- x= -36 y= 32 -98+198=100 --- x= -14 y= 18

4. List your values of x and y in an order x+ 19,30,41,52 -y -3,-10,-17,-24 -x -3,-14,-25,-16 y+ 12,18,25,32

5. Do you notice a pattern?

Yes, I can notice a pattern which consists in that the new term of the sequence of the “x” positive and negative you have to add 11. And “y” positive and negative I have to add -7 to the new term.

6. Can you make equations that describes how you get from one value of x and y to the next value?

x+

19,30,41,52

-y -3,-10,-17,-24
 * Un = 11n + 8 **
 * Un= -7n + 4 **

-x

-3,-14,-25,-36

y+
 * Un= 11n-47 **

32,25,18,11
 * Un= -7n + 39 **

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

Test: x+ Un= 11n + 8 Un= 11 (5) + 8 y- Un= 7n - 31 Un= 7 (5) - 31 7x+11y=100 7 (63) +11 (-31) =100
 * = 63 **
 * = - 31 **

8. Once you think you have the answer you need to write up the investigation explaining you’re working out at each stage.

Method for discovering the pattern and its formula:

1) Place in order the x+:

19,30,41,52

Find the difference between numbers in this case is: 11

Now Un=11n n represents any number we will test n with 1, so no it will be Un= 11(1) it will give you 11. find out how many more ore less to get to the first number in the list in this case: 19, you add 8. so it will be: Un= 11(1) + 8 the result will be x+ = 19

2) Put in order the values of -Y as the same order of the +X.

-24,-17,-10,-3

Repeat all step one but using this list of numbers

Un= -7(1) + 4 this will be -Y= -3

Now try both:

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

133+33=100

3) Repeat step 1 but using the list of numbers of negative X : -3,-14,-25,-36 and using the list of positive Y: 12,18,25,32.

9. This document must be uploaded to your Wiki portfolio that you started with Mrs. Garcia