I was helping my daughter with 8th grade algebra last night and came across an interesting problem. The problem was, write the equation for a line in the format y=mx + b that is perpendicular to the line y = 7 and contains the point (3, 1). In all of the other problems, you had to find the negative inverse of the slope for the given line and then use the point slope formula to solve into y = mx + b. It is obvious that the formula for the perpendicular line is x = 3. The question becomes, how do you show your work? The first thing that you would try to do is find the negative inverse of 0. What do you do after you try to divide by 0? I doubt that my answer of, “You can just look at the problem and tell what the answer is.” would be satisfactory to her teacher.
Maybe drawing a picture of the two lines (one horizontal, one vertical) would suffice?
Perhaps you could just write that since the line y=7 (or y=constant for that matter) is a horizantal line, any line perpendicular to it would not pass the “vertical line test” and therefor is not a function.
Perhaps you could simply set out the steps that your are intuitively taking:
- the line y=7 has a zero gradient, and is parallel to the x axis
- therefore any line perpendicular to it will have an undefined gradient (since one cannot divide by zero), making it, by definition, parallel to the y axis and thus of the form x = b
- if that line passes through the point (3,1), then it must have the equation x=3
It comes down to the fact that if a line is perpendicular to two or more lines, those lines are parallel, and vice versa.
Since y = 3 is parallel to y = 0, and x = 0 is perpendicular to y = 0, it follows that x = 0 is perpendicular to y = 3. So any line perpendicular to y = 3 must be parallel to x = 0.
Does that make sense?
When I was taking high school geometry we liked to appeal to the “obviousness theorem”. Sadly, when I got to college I discovered many examples of things that are both obvious and wrong.
No, it’s not a function, but it’s still a valid linear equation. Y is simply not a function of X. X can however be a function of a constant value C.
If she has learned that vertical lines have undefined slope, and that division by zero yields an undefined result, then it’s just a matter of answering the question, “What line goes through (3, 1) and has a slope which is undefined?” It’s just a matter of looking at it the other way. We know that the slope of the original line is zero. We know that the slope of a line perpendicular to this will be of the form C / 0, where C is any constant. We know that anything divided by zero yields an undefined result. We know that this result will be the slope of the line. We know that a vertical line is the only possible linear equation that can have an undefined slope. Once we have determined that the answer is a vertical line, we just have to determine which one.
We know that vertical lines have the form x = C (actually, if we’re being picky about having both X and Y coordinates so that we get a two-dimensional equation instead of a point on a number line, the form is x = 0y + c, but that’s a nitpick). We know that one possible solution to this equation is x = 3, y = 1 (from the given point. We can throw out the irrevelant y - value since no y appears in our equation.) Substituting 3 for X gives us 3 = C. Thus, since we now have the value for the “unknown” (but obvious) constant, substituting 3 back in the original equation gives us x = 3 (the obvious answer, as you said).
Well, on preview, I see that I’ve just spelled out in more words what Cunctator said. Oh well.
This sounds like a trick question. What your daughter is supposed to discover is that there is no solution of the form y = mx + b. One could look at this as either a mean trick or a good lesson. I tend to see it as the latter, since real-life applications of math can play similar mean tricks on you.
I think you’re probably right. It was problem number 53 out of 60 and so it should have been one of the tougher ones.
Thanks for all of the replies. I’m glad I wasn’t just missing something “obvious”. I asked Chelsea what her teacher said when they went over their homework in class. For some strange reason, this problem wasn’t covered. I suspect cowardice.
Well, obviously, the algorithm of just finding the negative inverse of the given line in order to find the slope of the perpendicular line breaks down when the given line has a zero slope, like for y = 7. The correct algorithm is:
(1) If the given line has slope not equal to zero, then find the negative inverse … etc.
(2) If the given line has zero slope, such as y = c, then the line perpendicular to it and passing through point (a, b) is x = a.
In other words, I would first correct the algorithm and then apply the appropriate part. And that’s how you show your work.