# How do you decide whether the relation 4x-7y=3 defines a function?

If you graph it, you can see that it is a straight linear function, as to be expected with an equation in the (rearranged) form $y = m x + c$. A good test when defining functions is the straight-line test: If you draw a straight vertical line down at any point through a graph of the function, you should only intersect with the function once - if you intersect with it more than once, it is not a function.