Finding the Equation of a Line

Date: 02/17/2005 at 21:24:00
From: Shelby
Subject: How do I figure out Equation of a Line Word Problems?

Find the equation of the line perpendicular to the graph of x - 3y = 6 
and with an x intercept of 6.

There are so many parts to it that I get steps mixed up!

I have already had this problem explained to me and still don't get 
how you get the correct answer!

Date: 02/17/2005 at 22:32:23
From: Doctor Peterson
Subject: Re: How do I figure out Equation of a Line Word Problems?

Hi, Shelby.

When you have a complex problem to solve, it helps to plan a strategy
to get from what you know to what you need.  Let's take inventory:
what do we know?

  1. We know the equation of a line.
  2. We know the x-intercept of the line we're supposed to find.

Now, where are we headed?

We want to write the equation of a line perpendicular to the given 
line, with the given x-intercept.

Next, we look at the goal, keeping in mind what supplies we are 
carrying with us, and ask what it would take to accomplish the goal:
what do we WISH we had?

Well, we know how to write the equation of a line if we know its slope
and y-intercept.  We don't have a y-intercept, but maybe we can get
one.  More important, the word "slope" reminds us of the word
"perpendicular": part of our goal is to make a line perpendicular to
the given one, and we know that the slope of a perpendicular line is
the negative reciprocal of the other.  So maybe it would be a good 
idea to find the slope of the line we were given. So let's make our 
first step to find the slope of that given line:

  3. Find the slope of the given line, by writing it in 
     slope-intercept form.

  4. Take the negative reciprocal of that to get the slope of the
     desired line.

That takes us a long way toward the goal; we'll have the slope and the
x-intercept of the line we want.  But it's unlikely you've learned the
"slope-x-intercept" form of a line; I myself know it exists, but  
haven't memorized it.  What other forms do we know?  Perhaps you know
the "point-slope" form of a line; if so, you can treat the x-intercept
as a point and apply that form:

  5. Use the x-intercept point (6,0) and the slope found in step 4 to
     write the equation of our line in point-slope form.

  6. Rewrite that equation in whatever form is desired, such as
     slope-intercept.

You may not have learned the point-slope form; if not, perhaps you 
have been taught an alternative way of finding a line through a point.
One way is to write the slope-intercept form, treating "b" as an
unknown; then you can take the known point and put its x and y into
the equation, and solve the resulting equation for b.  That gives you
the y-intercept, and you have the equation.

One way or another, you can solve the problem by taking one step at a
time, gathering the information you need for the final assault on the
goal.  In general, to find an equation of a line, you'll need to know
the slope and a point on the line.  Find the slope of the given line,
and then think about what you can use that for.  Mapping out a
strategy, rather than just blindly trying things, helps a lot in this
process.  I hope I've given you enough ideas to work out the rest for
yourself.

If you need more help, please write back and show me how far you got.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/