Is $y=mx$ direct or linear — or both?

Question

Suppose I have the equation $y=4x$ that I wish to graph.

Is this a direct relation or a linear relation?

I know it is a direct relation because the relationship between $x$ and $y$ is defined by the constant $4$ since it is, and makes $x$, a multiple of $y$.

However, it can also be a linear relationship. The general linear equation is $y=mx+c$ for reals $x, y, m, c$. Letting $c=0$ yields $y=mx$ which is a form obeyed by the equation $y=4x$ such that $m=4$.

So methinks it follows that $y=4x$ is both a direct relation and linear relation.

But the question in my book asks me to point out which graph out of three demonstrates the linear relation, the direct relation, and the non-linear relation (I have no trouble for the latter, however, because that is a quadratic). Each graph must correspond to each answer, and not more than one, implying that $y=4x$ is either direct or linear, albeit I disagree.

I could be wrong, though, hence why I have posted this question here for confirmation. Is $y=4x$ direct or linear? Or, for a more general question, is $y=mx$ direct or linear?

Thank you in advance.

Answer 1

$y=4x$ - and more generally, $y=mx$ - is indeed both a direct and linear relation. This sort of scenario follows a rule not unlike "all thumbs are fingers but not all fingers are thumbs:" all direct relations are linear, but not all linear relations are direct.

Direct relations are however a "stronger" implication, in the sense that more restrictions are placed on them, than linear. For a linear relation, for $y=mx+c$, any $c$ works, but for direct you must have $c=0$.

In such questions, usually, then, I would match the equations by those matching the stronger condition. So I would match $y=4x$ to a direct relation, the quadratic to the nonlinear, and then that - based on the nature of your question - leaves one equation and one option (linear).

I suppose the question would be clearer if the question stated "linear but not direct," but such is life. You wouldn't be wrong to say $y=4x$ is linear, technically, but this is just a consequence of how such questions are designed.