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Question: Write an equation for a line containing (-8,12) that is perpendicular to the line containing the points (3,2) and (-7,2)?

What I did (this is wrong): I found the slope for the second line, which was 0/-10. Then I changed it to be perpendicular, which was -10/0. Then I found the y-intercept by plugging in (-8,12) into the slope-intercept form and got b= -7. So, the slope intercept is y= -10/0x + 12.

This was obviously wrong, but I didn't know what to do. So, if someone can please help me, that would be great.

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    $\begingroup$ For some basic information about writing math at this site see e.g. here, here, here and here. $\endgroup$ – user137731 Nov 16 '14 at 19:55
  • $\begingroup$ First off, if you're trying to divide by $0$, you should realize you did something wrong. $\endgroup$ – user137731 Nov 16 '14 at 19:56
  • $\begingroup$ I know, but what do I do then $\endgroup$ – joe Nov 16 '14 at 19:57
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This seems like a good point to discuss a little bit what the slope of a line means. Consider the line $$-2x+y=6\tag{1}$$ (this is called standard form, but feel free to add $2x$ from both sides if you prefer point-slope form).

enter image description here

Let's try to interpret this physically. Let's let $x$ denote the time, in seconds, from the point where we started our stopwatch and $y$ will denote the distance, in meters, a runner has traveled in that time. So for instance, we see that the runner was already $6$ meters from the starting position when we started our stopwatch because $$-2(0) + y=6 \\ \implies y=6$$ When the stopwatch reads $2$ seconds, the runner is at $$-2(2)+y=6 \\ \implies y=10\text{ meters}$$ Likewise, the time the stopwatch reads when the runner has run $100$ meters is $$-2x + 100 = 6 \\ \implies x=47\text{ seconds}$$ We can also use this equation to backtrack a bit and figure out when the runner actually started the race -- as opposed to when we started the stopwatch. This is when $y=0$: $$-2x+(0) = 6 \\ \implies x=-3$$ So the runner started $3$ seconds before we started the stopwatch. Hence, if this was a 100 m race, then it took the runner $47+3=50$ sec to run the whole race.

As you can see, we can get a lot of information from equation $(1)$. But what about the speed of the runner? Can we get that information? Yes we can. Recall that $\text{speed} = \dfrac{\text{distance}}{\text{time}}$. So the speed of the runner during the 100 m race was $$\text{speed} = \frac{100 \text{ m}}{50\text{ sec}} = 2 \text{ m/s}$$

But notice that $\dfrac{\text{distance}}{\text{time}}$ is exactly the same as $\dfrac{\Delta y}{\Delta x} = \dfrac{\text{rise}}{\text{run}}$ in this case. Hence the speed is just the slope of the line.


Let's try think to about all lines in this way -- they represent a runner running a race. Then a greater slope means a faster runner. A negative slope means the runner is running the wrong direction! A slope of $0$ means the runner is protesting the race and has decided not to run at all (he's standing still). There's one type of line where this analogy doesn't work, though. A vertical line. When a line is vertical how fast is the runner running? It must be something like infinitely fast because in $0$ time he moves some nonzero distance. This doesn't really make sense and because of this, it doesn't really make sense to talk about the slope of a vertical line -- we say the slope of a vertical line is undefined.


This is exactly the problem you're having in your exercise. The line through $(3,2)$ and $(-7,2)$ is horizontal. It's the line $y=2$. So what type of line is perpendicular to a horizontal line? A vertical line. So you won't be able to get a slope from it -- you'll just end up dividing by $0$. However, we can still use an equation (in standard form) to represent vertical lines. For instance, the line through $(1,1)$ and $(1,6)$ is $x=1$. The line through $(6,3)$ and $(6,-2)$ is $x=6$. What should the equation of your line be?

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As per Bye_World's comment:

First off, if you're trying to divide by 0 , you should realize you did something wrong.

No, not really. Note that the slope of the other line is $0$. The slope of this line is $\frac{1}{0}$. This means that this line is vertical, as it should be, given that it's perpendicular to a horizontal line.

All vertical lines can be written as $x=c$, where $c$ is a constant. All you have to do here is figure out what $c$ is.

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  • $\begingroup$ I understand the heuristic, but I have a problem with someone saying "the slope of the line is infinity". $\endgroup$ – user137731 Nov 16 '14 at 20:01
  • $\begingroup$ @Bye_World What would you prefer I said? It's ugly, but there's no way around it. $\endgroup$ – HDE 226868 Nov 16 '14 at 20:01
  • $\begingroup$ Something like what I wrote, I guess $\endgroup$ – user137731 Nov 16 '14 at 20:02
  • $\begingroup$ @Bye_World So just leave that part out? $\endgroup$ – HDE 226868 Nov 16 '14 at 20:03
  • $\begingroup$ @Bye_World Made the edit. $\endgroup$ – HDE 226868 Nov 16 '14 at 20:04

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