Expert: Chanda Walker - 1/3/2009

Question
QUESTION: Hello! Recently, I have found myself struggling with many questions in my math class related to slopes and proofs. I understand finding slope of a line, but it's the questions that ask for you to find lines perpendicular to others. For example, one of them is:
What is the slope of a line perpendicular to the line whose equation is 5x+3y=8?
I guess i just don't know how to approach the problem. I knpw they say you have to multiply the gradient or something? But i just don't get it.
Help would be greatly appreciated! Thanks.

ANSWER: The slope of a line perpendicular to a line with slope given by m is

-1/m

It is pretty easy to do.  Just invert the original line's slope and make it negative.

For the problem you mentioned:
y = -5/3x + 8/3

The slope of the line perpendicular would be 3/5.

Let me know if you need more explanation.


---------- FOLLOW-UP ----------

QUESTION: Thank you very much! I believe i understand it a little bit better to a certain degree. But, i do have one more follow-up question.
So, first of all i would solve the equation 5x+3y=8 out for y, to get: y= 8/3 - 5/3x.
I get where you are finding the 3/5 as the perpedicular line, you took theinverse of the -5/3. But, will i always take the inverse of the fraction next to the x variable? And never the other fraction which in this case is 8/3?

A second question: When they ask for a bisecting line or parallel line will i approach the question with the same method?


ANSWER: You will ALWAYS use the coefficient (in this case it was a fraction) of the x term.

Let me back up if you will allow.

y = m x + b

This is the general form for EVERY line.  The slope of the line is given by m.  If m is 1, the line rises along 45 deg with respect to the x-axis.  A larger slope rises faster; a smaller slope rises slower; negative slope goes down.

When you invert and make negative, you have found the line perpendicular to the original line.  The m controls the slope.

The b moves the line up in down.  b is called the intercept.  It is where the line crosses the y-axis.  You can "see" this because if you let x=0 in the equation, you will find what y is on the axis.  So changing the b only moves the line, it doesn't rotate it.

So yes, the coefficient of the x term always is in control of the slope and so that is the only you one you must change to make it perpendicular.

The second question:

To find the slope of EVERY line perpendicular do just what I've shown you.  To find the SPECIFIC line that passes through a point, you have to adjust the b term.

When you want to find the b term for a line that bisects the original two points given (which is a VERY common problem) all you have to do is place the bisecting point (x_bi, y_bi) into the general form for a line and solve for b.  I'll show you how but remember, you should already know m and I'm going to leave it as m in my equation but it will be a number for you.  Here goes!

y = m x + b   (general form)

y_bi = m x_bi + b

b = y_bi - m x_bi

That is it.  You have m now.  You have b now.  Put those back into the general form and you're done.

Please feel free to ask a specific problem and I'll walk you through it.  Using variables all over the place can be a little confusing.

Or if you look back through a few of the problems I've worked for others here, you'll find an example.  Slope and intercept problems are popular here.
Good luck!

---------- FOLLOW-UP ----------

QUESTION: Hm. Your explanation about finding the perpendicular line to antoher is much clearer for me, and i thank you for that! I'm sure it will help me much more. But, I guess what I'm confused about is finding a line that passes through a point, or is parallel. I apologize, for your explanation was very thorough. But may i ask; what does the y_bi mean? the underscore? I assumed it meant a comma, but i wasn't quite sure. I've solved a problem which was...
   WHAT IS THE EQUATION OF A LINE THAT PASSES THROUGH THE POINT (-3,11) AND IS PARALLEL TO THE LINE WHOSE EQUATION IS 2X-Y=4.
i solved it out by taking 2x-y=4 and solving for y to get 2x-4. I then entered the line's equation into my calculator, and also the multiple choice's. I paired it up with the equation y=2x-5, as it they were parallel. Now that i think about it, i guess i assumed it passed through the points of (-3,11), which i should probably check. I know that I took a lot of short cuts to solve it, and i know there IS a mathematical formula. If you could please just use this specific example to explain it to me, I'd really appreciate it! Thanks again! :)


Answer
To answer about the y_bi:

Sorry this wasn't clear enough!

A common algebra problem will give two points (x_1, y_1) and (x_2, y_2) and ask for the line perpendicular to this segment that bisects the segment.

I use the _ to act as subscripts.  So the first point is located at x_1 and y_1.  The second point at x_2 and y_2.  Well it is easy to find the point that bisects that segment (just average the x coordinates and then average the y coordinates).  I was using the notation (x_bi, y_bi) to denote the coordinates of the bisector.  Bi for bisector.  Does that makes sense?

If they ask you to make the line go through a different point, just put the coordinate of that other point in for the bisector coordinate.  Your problem uses (-3, 11) so I'll do that example next.

For the problem of parallel lines:

The equation for your original line looks like this.

2x-y = 4

y = 2x - 4   (just like you did)

therefore m = 2

Your new line looks like this so far

y = 2 x + b

Now just use the point given in the place of y and x and solve for b.

11 = 2 (-3) + b

b = 11 + 6
b = 17

So your new line is:

y = 2 x + 17

I'm happy to help you work on this as long as you need it.  Keep asking if it isn't clear.  Tutoring over the internet with text isn't the easiest way to do this but you seem like you are trying very hard so it is very rewarding to me to try to help.