Is it possible to solve for two unknowns from one equation?

Is it possible to solve for two unknowns using only one equation?

For example:

$x+3y=32$

Where $x$ and $y$ are integers.

Thanks :)

Solution 1:

The answer to the original question 'Is it possible to solve for two variables related by single equation' is, YES!

Here is an example. Solve for two unknown real numbers $x$ and $y$ given a single equation $$x+\sqrt{y-1}=\sqrt[4]{1-y}$$ The only real solutions are $x=0,y=1$

Solution As $x,y$ are real numbers, either $y-1\ge0$ or $1-y \le0.$

Assume that $y$ is strictly greater than $1$, then the LHS of the equation is a real number, whereas, the RHS will have non-zero complex part. A similar argument is also true when the value of $y$ is strictly less than $1$. Therefore, $y = 1$ is the only possible solution. Indeed, $x = 1$ and $y = 0$, is a solution, and the only one.

This may be considered as a hack, since apart from the equation, we have two constraints, i.e., $x$ and $y$ are Real. There are complex solutions to the equation.

Solution 2:

Not uniquely. You can solve for $x$ with $x = 32 - 3y$, and if you choose any $y$, you'll have an $x$ so the corresponding $(x, y)$ pair is a solution. Conversely, you can solve for y in terms of $x$, and then use (any? or just integer $x$ so that $y$ is an integer -- you filed this under "diophantine equations") $x$ choices to get $y$s for $(x, y)$ solution pairs.