I have seen few math problems online, about solving one equation with two unknowns, (which is not possible as the number of equations should match the number of unknowns), but I thought is there any way to prove that LHS = RHS for some integer (...,-1,0,1,2...) numbers without substituting values like in trail and error method.
Suppose, I have an equation say 3x + 5y = 22, by looking at it for few seconds, I can substitute the values x = 4 and y = 2, but I want to know if there is any way to know that there exist a solution for the equation such that x and y are integers.
P.S. I don't know what to tag please edit the tags if found any better ones.
The slope of your line is $-\frac{3}{5}$. This is to say that for every $5$ units you travel in the $x$-direction, you move $-3$ units in the y-direction.
Since you know that $x=4, y=2$ is a solution, then you can move right $5$ units and down $3$ units to get another solution. You can do this infinitely many times. In addition, you can go in the opposite direction (left and up) and get infinitely many integer points as well.
Then the general solution where both $x$ and $y$ are integers is:
$$(4+5t, 2-3t)$$
where $t$ can be any integer value.