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For the given pair (x, y) of positive integers, 7x + 3y = 123. How many integral pairs of x and y satisfy the given condition?
- a)2
- b)3
- c)4
- d)5
Correct answer is option 'D'. Can you explain this answer?
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For the given pair (x, y) of positive integers, 7x + 3y = 123. How man...
Solution:
To find the integral pairs of x and y that satisfy the given condition, we need to solve the equation 7x - 3y = 123 for positive integer values of x and y.
To begin solving this equation, we can rewrite it as 7x = 3y + 123.
Since x and y are positive integers, we can start by finding the smallest positive value of y that satisfies the equation. We can do this by setting y = 1 and solving for x:
7x = 3(1) + 123
7x = 3 + 123
7x = 126
x = 18
So, one integral pair of x and y that satisfies the equation is (18, 1).
To find more integral pairs, we can increment the value of y and solve for x again.
y = 2:
7x = 3(2) + 123
7x = 6 + 123
7x = 129
x = 18.42857 (not an integer)
y = 3:
7x = 3(3) + 123
7x = 9 + 123
7x = 132
x = 18.85714 (not an integer)
y = 4:
7x = 3(4) + 123
7x = 12 + 123
7x = 135
x = 19.28571 (not an integer)
We can see that for y = 2, 3, and 4, the value of x is not an integer. Therefore, there are no more integral pairs of x and y that satisfy the equation.
Hence, the correct answer is option D) 5, as there is one integral pair (18, 1) that satisfies the equation.
To find the integral pairs of x and y that satisfy the given condition, we need to solve the equation 7x - 3y = 123 for positive integer values of x and y.
To begin solving this equation, we can rewrite it as 7x = 3y + 123.
Since x and y are positive integers, we can start by finding the smallest positive value of y that satisfies the equation. We can do this by setting y = 1 and solving for x:
7x = 3(1) + 123
7x = 3 + 123
7x = 126
x = 18
So, one integral pair of x and y that satisfies the equation is (18, 1).
To find more integral pairs, we can increment the value of y and solve for x again.
y = 2:
7x = 3(2) + 123
7x = 6 + 123
7x = 129
x = 18.42857 (not an integer)
y = 3:
7x = 3(3) + 123
7x = 9 + 123
7x = 132
x = 18.85714 (not an integer)
y = 4:
7x = 3(4) + 123
7x = 12 + 123
7x = 135
x = 19.28571 (not an integer)
We can see that for y = 2, 3, and 4, the value of x is not an integer. Therefore, there are no more integral pairs of x and y that satisfy the equation.
Hence, the correct answer is option D) 5, as there is one integral pair (18, 1) that satisfies the equation.
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