Integer solution to the equation 3x+4=7y

Question

An integer solution to the equation 3x + 4 = 7y

is an ordered pair of integers (x, y) that satisfies the equation. For example, (1, 1) is one such solution. Write the set of all integer solutions to the equation 3x + 4 = 7y

 in set builder notation.

Answer 1

Solving for y : y = (3/7)x + (4/7)

we can see more easily the solutions are (1,1), (8,4), (15,7), ....

Starting with x=1, every time you increase (or decrease) x by 7, you find another integer solution.

Algebraically, the solutions are x = 7k+1 where k = 0, ±1, ±2, ...

In set bulider notation, solutions are { x | x = 7k+1 ; |k| = 0,1,2,3,... }