Since x and y both can take negative integers value less than 1000 and coefficients of both x and y are positive, in the given equation x and y would take opposite sign such that ‘4x + 7y’ equals 3 always.
Also, we have
-1000 < x < 1000 and -1000 < y < 1000 OR -999 ≤ x ≤ 999 and -999 ≤ y ≤ 999 (under given conditions)
To find combination of x and y satisfying 4x + 7y = 3.
Here if x = -1 then y = 1. Now the integer values of both x and y would vary according to coefficients of one another i.e. x would vary by magnitude of 7 integer values and y would vary by magnitude of 4 integer values. Additionally, both x ≠ 0 and y ≠ 0.
Thus, x = 13, 6, -1, -8, -15 and so on and y = -7, -3, 1, 5, 9 and so on.
However, looking at the equation 4x + 7y = 3 and the range of values of both x and y it can be observed that y would have more values in the given range than x. So the number of values of x would be sufficient to find the combination of values x and y.
Finding the possible minimum value of x -999 does not satisfy the given equation since \(\frac{(-999-(-1))}{7}\) leaves a remainder o 4. Hence the least value of x is -995.
Similarly, the possible maximum value of x 999 does not satisfy the given equation since \(\frac{(999-6)}{7)}\) leaves a remainder o 6. Hence largest value of x is 993.
Calculating number of values x' which x can take between -995 and 993 both inclusive gives –
\(x' = \frac{(993-(-995))}{7}\)
\(x' = 284\)
Answer (A).
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