How to find the equation of a line perpendicular to another?

**Ashir** · September 22nd 2012, 08:25 AM

How does y+2=0 rearrange to y=-2? And how do you find a perpendicular line (with a given coordinate)?
a = 0,1
b = 10,6
Equation = y=1/2x+1

Find the equation of the line perpendicular to AB passing through B.

Please explain all in a 'step-to-step' fashion in the simplest way possible including any rearranging equations etc.
I know the answer (y=-2x+26) but, other than the negative reciprocal, I'm not sure how to get it. My answer on the exam was 'y=-2x+6' so I went wrong at the end.
Thanks a lot!

**MarkFL** · September 22nd 2012, 08:40 AM

How does y+2=0 rearrange to y=-2?

If you are given:

$y+2=0$

and you want to solve for $y$ , then since 2 is being added to $y$ , we may isolate $y$ by subtracting 2 from both sides:

$y+2-2=0-2$

$y=-2$

And how do you find a perpendicular line (with a given coordinate)?

You are given the points:

$A(0,1),B(10,6)$

The perpendicular slope is found by:

$-\frac{\Delta x}{\Delta y}=-\frac{10-0}{6-1}=-\frac{10}{5}=-2$

We now have the slope of the line and the point it passes through, so using the point-slope formula, we find the equation of the line is:

$y-6=-2(x-10)$

$y=-2x+26$

**Ashir** · September 22nd 2012, 08:51 AM

Ah, understood, thanks.

I can't catch on the last bit. Is there an easier way or could you explain how you got to that?

**MarkFL** · September 22nd 2012, 09:02 AM

$y-6=-2(x-10)$

$y-6=(-2)(x)-(-2)(10)$

$y-6=-2x+20$

$y-6+6=-2x+20+6$

$y=-2x+26$

**Ashir** · September 22nd 2012, 09:13 AM

I'm really sorry but I still can't understand it. Where did -6 come from? Where did -10 come from? Why was the '-2' moved?
Could you explain the format of the above expressions please? And what is this 'point-slope formula'? y=mx+c or?

**MarkFL** · September 22nd 2012, 09:21 AM

The point-slope formula states that given the point $(x_1,y_1)$ on a line and the slope $m$ , then the equation of the line is:

$y-y_1=m(x-x_1)$

The -2, which is the slope, was distributed to both terms within the parentheses.

**Ashir** · September 22nd 2012, 09:31 AM

I understand now, but is there another way to do it? I don't think I'll remember that. I have trouble remembering that a line sloping to the left is +ve and to the right -ve (not even sure that's correct)
Or is there any way to relate that formula to y=mx+c? Or is there another formula altogether? I know how to get the slope, so is there another way to get the y intercept (c)?

**MarkFL** · September 22nd 2012, 10:20 AM

Yes, use the formula for slope with the points (10,6) and (0,c):

$-2=\frac{c-6}{0-10}$

$c-6=20$

$c=26$

**Ashir** · September 23rd 2012, 01:46 AM

My friend taught me an easier way. Substitute 'y' and 'x' with the coords of the point it passes through (B)

6=-2(10)+C
6=-20+C
26=C

Thanks for the help though!

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