Cal+1+Paper+6+Investigation

Investigation:

First of all I will find out pairs of numbers which satisfy the equation (7x + 11y=100), which will help me to find the formulas.

First of all I will plot all the different sums that satisfy the equation mentioned above. I will also plot the x and y values, this would help me find the formula after more evidences.


 * 1) 133 + -33 = 100 erterfgdhgdfghdfghdfghdfghdf x= 19 y= -3
 * 2) -98 + 198 = 100 fdgsdfgsdfgsdfgsdfgsdfgsdfgsk x= -14 y= 18
 * 3) 56 + 44 = 100 gsdfgsdfgsdfgsdfgsdfgsdfgfdsg x= 8 y= 4
 * 4) -21 + 121 = 100 gfhdfghdfghdfghddfghdfghdfgh x= -3 y= 11
 * 5) 210 + -110 = 100 gdhdfghdfghdfghdfghdfghdfgh x= 30 y= -10
 * 6) -175 + 275 = 100 sdfgsdfgsfgsdfgsdfgsdfgsdfds x= -25 y= 25

Above we can see some of the possible sums which satisfy the formulae: (7x + 11y=100). I will take example number 3 and number 5 to prove that they work.

Example Number 3: (7 x 8) + (11 x 4) = 100 56 + 44 = 100

Example Number 5: (7 x 30) + (11 x -10) = 100 210 + -110 = 100

With these two examples I know that my sums are correct because they add to 100 using the same equation from the beginning and now I can continue to the next step to find the formulae.

Now I will write the x values in order, from lowest to biggest so I can see a pattern and create the formulae for the x value side.

X Values in order:

30, 19, 8, -3, -14, -25

Using this 6 different x values I noticed a pattern. The pattern is that it goes decreasing 11 each time. From 30 it goes to 19, this means it decreased 11. From 19 it goes to 8, this means the same it decreased 11. From 8 it went to -3, this means it decreased 11 from 8 to make -3. This pattern will go forever.

Knowing that there is a certain pattern on this sequence we should use the nth term to find the x value equation.

Nth Term: -11n + 41

Now I will test my nth term to see if it’s correct.

(-11 x 1) = -11 + 41 = **30**

(-11 x 2) = -22 + 41 = **19**

(-11 x 3) = -33 + 41 = **8**

(-11 x 4) = -44 + 41 = **-3**

(-11 x 5) = -55 + 41 = **-14**

(-11 x 6) = -66 + 41 = **-25**

I have now proved that my nth term is correct because all the answers correspond to the x values that I found. Knowing it works I will start making the equation. So far it would look like this: **7(-11n+41) + 11 = 100.**

We are still missing the y nth term so to complete the missing part of the equation. I will do exactly the same as I did on the x values. I will start putting the values in order.

Y Values in order:

-10, -3, 4, 11, 18, 25

Using this 6 different **y** values I noticed a pattern. The pattern is that it goes increasing 7 each time. From -10 it went to -3, this means it increased 7 from -10 making -3. From -3 it went to 4, this means the same it increased 7. From 4 it went to 11, this means it increased 7 from 4 to make 11. This pattern will continue forever.

Now that I know that there is a pattern I will use the nth term to find the **y** value equation so we can complete the entire formulae.

Nth Term: 7n – 17

Now I will find out if my nth term is correct.

(7 x 1) = 7 - 17 = **-10**

(7 x 2) = 14 - 17 = **-3**

(7 x 3) = 21 - 17 = **4**

(7 x 4) = 28 - 17 = **11**

(7 x 5) = 35 - 17 = **18**

(7 x 6) = 42 - 17 = **25**

Watching at my results I can say that my nth term is correct because it gave me the correct numbers that are on the **y** values. Now I can finally add the last part of the equation and test it to see if it works.

So the finished equation would be:


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

So having the equation **7(-11n+41) + 11(7n-17) = 100** we can test if it is correct.

Let’s do 2 examples, one with **n** being positive and another one with **n** being negative.

First Example using n = 7


 * ( ** -11 x 7 = -77 + 41 = -36 **x 7=** **-252)** **+ (**7 x 7 = 49 – 17 = 32 **x 11 = 352)**


 * _ -252 + 352 = 100 **

Second Example using n = -3


 * ( ** -11 x -3 = 33 + 41 = 74 **x 7** **= 518)** **+ (**7 x -3 = -21 – 17 = -38 **x 11 = -418)**


 * 518 + -418 = 100 **

I have finally proved that my equation is successful, it **works** for any **positive** or **negative** integer, and all the addition or subtraction taking place on this equiation will always be **100**.