Example

Find two solutions of $2x-y=9$.

Solution

This equation is a rather "nice" equation involving integer coefficients. It is not difficult to simply notice or observe some solutions. For example, $(x,y)=(5,1)$ is a solution, and so is $(x,y)=(0,-9)$. You might be able to just notice that these ordered pairs satisfy the equation without doing any real work.


Using powers of observation isn't guaranteed to find solutions---observation depends entirely on you being able to notice solutions. Here is a method that will always work: first choose a variable from the equation, replace it with any number you'd like, and solve for the remaining variable. For example...

First solution: I choose the variable $x$, and I replace it with $x=7$:

or

or

The point $(7,5)$ is a solution.

Second solution: I choose the variable $y$, and I replace it with $y=10$:

or

or

The point $(19/2,10)$ is a solution.