# Lines in coordinate plane

A question from an ACT math test:

What are all the real value(s) of $$m$$, if any, such that any line through the points $$(3, 7)$$ and $$(3, m)$$ will be vertical when graphed in the standard $$(x,y)$$ coordinate plane?

I do not understand what the question is asking when it says 'a line will be vertical', but vertical to what - $$x$$ axis or $$y$$ axis?

• Try $(3,8)$ and then think about it. Oct 3 '20 at 4:58
• In the $x,y$ coordinate system, where the $x$-axis is regarded as horizontal, and the $y$-axis is regarded as vertical, a line will be regarded as vertical $\iff$ the line is parallel to the $y$-axis, which will be $\iff$ the $x$ coordinate is constant on the entire line. Oct 3 '20 at 5:15

A vertical line here and in most any context involving the Cartesian Coordinate system you'd encounter on a test is line "up-and-down" the page, i.e. it is parallel to the $$y$$-axis. (This is pretty much the standard linguistic meaning of vertical too.)
To think about answering this question recall that any vertical line in the coordinate plane is described by $$x = k$$ for some $$k$$ a real number. So let's consider $$x = 2$$, it is the locus (meaning collection) of all points $$(x,y)$$ in the plane for which $$x=2$$ and $$y$$ is anything we like. Thus $$(2,1)$$ lies on this line, so too does $$(2,-10)$$ and $$(2,5643)$$. I strongly encourage you to plot (using graph paper can always be helpful) some of these points and see what you get.
With the above in mind then think about what real numbers $$m$$, in the $$x$$-coordinate of $$(3,m)$$, will give you a vertical line passing through $$(3,7)$$? (And graph this point too, that's really often the key to coordinate problems, let your mind actually have some geometry to work with in front of you.).