Basic question about $y=mx$

Question

The graph of equation $y=mx$ is a straight line...

a) Parallel to the $x$-axis

b) Parallel to the $y$-axis

c) Passing through the origin

d) That coincides with the $x$-axis

This was a question in a math quiz which I don't know the answer to. Can anybody help me?

Answer 1

If $y=mx$, plugging in $x=0$ will result in $y=m\cdot 0=0$. Hence, the line passes through the origin, which is the point $(0,0)$.

Let's analyze the other answers and show that generally they are not true.

a) For a line to be parallel to the x-axis, y must be constant. If the equation of the line is $y=mx$, this happens just for $m=0$: otherwise $y$ would vary proportionally to $x$.

b) An equation of the type $y=mx$ can never represent a vertical line, because the slope of a vertical line is not finite.

d) The x-axis' equation is $y=0$, so $y=mx$ will coincide with the x-axis if and only if $m=0$.

Answer 2

(a) True if and only if $m=0$.

(b) Never true.

(c) Always true (since $x=0$ implies $y=0$).

(d) True if and only if $m=0$.

Answer 3

A: depends on $\boldsymbol m$. For a straight line parallel to $x$ axis, the equation is $y=$ some constant. So it's false in general, true if $m=0$.

B: false. The equation should be $x=$ some constant.

C: true. Indeed , as the origin has coordinates $(0,0)$, the equation is satisfied, whatever $m$.

D: depends on $\boldsymbol m$. $m$ must be $0$.

Answer 4

I think the main problem here is in how you've phrased the question. You have almost certainly left out that $x$ ranges over all of $\mathbb{R}$ while $m$ is a fixed value in $\mathbb{R}$. The purpose of the question is likely to test your understanding of what is true of $y=mx$ for all $m$ (where it is implicitly assumed that you are considering $x$ to range over all of $\mathbb{R}$). As other answers have pointed out, only option (c) is going to be true all of the time for $y=mx$, irrespective of the values chosen for $m$ (this is because $y=mx$ passes through the origin $(0,0)$ whenever $x=0$, regardless of the value of $m$).

I cannot see why else someone would phrase a multiple choice question like this, where presumably only one answer is correct. Of course, it would be entirely different for you to have to characterize when all of the other answer choices may be correct. Other answerers have given ways to tackle those possibilities.