How many ordered pairs (x, y) of counting numbers (x = 1, 2, 3, · · · , and y = 1, 2, 3, · · ·) satisfy the equation x + 2y = 100?
x+2y=100
2y=100-x Divide with 2
y=(100-x)/2
y=50-x/2
x=2 y=50-2/2=50-1=49
x+2y=2+2*49=2+98=100
x=4 y=50-4/2=50-2=48
x+2y=4+2*48=4+96=100
x=6 y=50-6/2=50-3=47
x+2y=6+2*47=6+94=100
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x=44 y=50-44/2=50-22=28
x+2y=44+2*28=44+56=100
x=46 y=50-46/2=50-23=27
x+2y=46+2*27=46+54=100
x=48 y=50-48/2=50-24=26
x+2y=48+2*26=48+52=100
x=(2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48)
y=(49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26)
24 pairs
thank you for the response;
why is x is limited to 48 , and y to 26;
why can't we take x upto 98?
x=98, y = 1
x=96, y = 2
x=94, y = 3
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..
..
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x=6, y = 47
x=4, y = 48
x=2, y = 49
with this it comes to 49 pairs....is this correct?
50=n/2+26
To find the number of ordered pairs (x, y) of counting numbers that satisfy the equation x + 2y = 100, we can use a systematic approach. Since x and y are counting numbers, we need to find the values of x and y that satisfy the equation and fall within the range of counting numbers.
Let's start by examining the range of possible values for x and y. Since x + 2y = 100, we know that both x and y must be less than or equal to 100 since their sum is 100. Similarly, x and y cannot be negative, so their minimum possible values are 1.
We can set up a loop to check all valid values of x and y to see which ones satisfy the equation.
Here's some Python code that demonstrates this approach:
count = 0 # Variable to store the count of valid ordered pairs
for x in range(1, 101):
for y in range(1, 51): # The range of y is limited to 1 to 50 since 2y should not exceed 100
if x + 2*y == 100:
count += 1
print(count)
By running this code, we iterate through all valid values of x and y and check if they satisfy the equation x + 2y = 100. The count variable keeps track of the number of valid ordered pairs.
The output of this code will be the number of ordered pairs (x, y) of counting numbers that satisfy the equation x + 2y = 100.