Download Write 100 as the sum of two integers, one divisible by 7 and the

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.
Positive integers
x
y
7
93
14
86
21
79
28
72
35
65
42
58
49
51
54
44
63
37
70
30
The multiples of 7 are laid at the x axis stating from 7 up to 70 and the numbers at the y axis
are the numbers which will add up to 100. In other words x and y = 100. And to find the
satisfying condition you have to find a number in the y axis which is a multiple of 11.
As for the results there is only 1 pair of positive integers which is 56 and 44
Equal to :
X= 8
Y=4
7x+11y=100
7(8) +11(4) =100
56+44=100
2. Now try negative integers for x with positive integers for y and vice-versa.
Negative integers
x
y
-7
107
-14
114
-21
121
-28
128
-35
135
-42
142
-49
149
-54
154
-63
163
-70
170
-77
177
-84
184
-91
191
-98
198
-105
205
-112
212
For negative integers I did the same thing as in positive integers which is to lay in x axis the
multiple of 7 but in negative numbers starting from -7 up to -112 and in y axis the number
which will add up to 100. (-x+y=100). And as in positive integers I have to find a number which
will satisfy both conditions in x axis a multiple of 7 and in y a multiple of 11.
I found 2 pair of negative integers which are -21 & 121, -98 & 198
X= -3
7x+11y=100
Y= 11
7(-3)+11(11)=100
-21+121=100
X= -14
Y= 18
7x+11y=100
7(-14)+11(18)=100
-98+198=100
3. 7x+11y=100 solutions (positive and negative integers)
x
y
8
4
-3
11
-14
18
In this table at the x axis are the numbers which are multiplied to make the integers of 7 and
in y axis integers of 11.
4. Do you notice a pattern?
Throughout the investigation I noticed a pattern which is that in x (multiples of 7) goes
subtracting 11 each time starting from 8 (7 x 8) and in y (multiple of 11) it goes adding 7
starting from 4. (11 x 4)
5. Can you make equations that describes how you get from one value of x and y to
the next value?
Equations:
These are the equation I found for nth term of x and nth term of y
X= -11n+19
Y= 7n-3
7x+11y=100
7(-11n+19) +11(7n-3) =100
6. If you can, test to see if your equations work for several other values of x and y
Checking equations
X= -11(7)+19 = -58
7x+11y=100
Y= 7(7)-3 = 46
7(-58)+11(46)=100
-406+506=100
X= -11(10)+19 = -91
7x+11y=100
Y= 7(10)-3 = 67
7(-91)+11(67)= 100
-637+737=100