Problem Solving (PS) | Re: How many integer values of x and y satisfy the expression 4x + 7y = 3

first we need to find out a solution which is valid.

4x + 7y = 3
x = -1, y = 1 is a valid solution.

The valid values of x will be in an Arithmetic Progression where the common difference is the coefficient of y.
y = (3-4x)/7

=> Valid values of x are = ... -15, -8, -1, 6, 13, 20...

We are given |x| < 1000
=> Valid values of x are = -995, -988... -15, -8, -1, 6, 13, 20.... 986, 993 (I substituted x with a few numbers around 990 & -990 and checked the resultant value of (3- 4x)