Question

Write an equation of the line that is perpendicular to the line 2x + 5y = -15 and contains the point (-8,3).

Another one of these @Littlebird I'm going to try do it and see if I get it ;P

Answer 1

First I convert the standard form of 2x + 5x = -15 into y intercept form.

I get: y = -2/5x - 3

Then I input the point (-8,3)

3 = -2/5(-8) + b

Simplified

3 = 3 1/5 + b

Subtracted 3 1/5 from both sides

Got: -1/5 = b

@Littlebird @DanJS Is this correct?

Answer 2

@wio @nincompoop @Jhannybean @Jesstho.-.

Answer 3

hi

Answer 4

You kept the slope the same

Answer 5

looks correct.

Answer 6

So you are using parallel line

Answer 7

OH YES, I knew I forgot something

Answer 8

Parallel slope to -2/5 is 2 1/5 right?

Answer 9

negative inverse

Answer 10

\[2x + 5y\] will be perpendicular to \[
-5x+2y
\]

Answer 11

Oh, it asks for the perpendicular of the line, not the slope...

Answer 12

So I have to negative inverse everything o-o

Answer 13

Basically, when you have: \[
ax+by=c
\]Then the perpendicular line will be a form like: \[
-bx+ay
\]Does that make sense?

Answer 14

In our case, we have: \[
-5x+2y = \ldots
\]To find out what \(\ldots\) is, we have to plug in the point they gave us.

Answer 15

(-8,3)

Answer 16

So we get: \[
-5(-8)+2(3)=\ldots
\]

Answer 17

How did you get -5x + 2y from 2x + 5y = -15

Did you switch the variables around?

Answer 18

I thought you were supposed to simplify the line from 2x + 5y = -15 into slope intercept form

Answer 19

Well, the easiest way to explain it is with vectors, but you don't know vectors yet.

Answer 20

I mean, you could even use \(5x-2y\) and it would work as well

Answer 21

Since you're just multiplying by \(-1\).

Answer 22

Another way to explain it, other than vectors, is through rotations.

Answer 23

do you remember how to rotate a point?

Answer 24

no

Answer 25

\((x,y)\) rotated by \(90^\circ\) is going to be \((-y,x)\).

Answer 26

Basically \[
\color{red}{(x,y)}\to \color{blue}{(-y,x)}
\]For our line: \[
2\color{red}x + 5\color{red}y = -15 \to 2(\color{blue}{-y})+5(\color{blue}{x})=-15
\]This simplifies to:\[
5x-2y=-15
\]However, this line doesn't necessarily go through our point, which is why I dropped the \(-15\).

Answer 27

Anyway, since: \[
-5(-8)+2(3)=-40+6=-34
\]Our line is given by: \[
-5x+2y=-34
\]Alternatively:\[
5x-2y=34
\]Anyway, if you want, you can put it into slope intercept form.

Answer 28

@SarahLove<3 Does it make sense?

Answer 29

Yeah, the way it was taught for us was to use the point slope form, just remembered that.

Answer 30

Your way makes sense, perpendicular is basically a line that forms a right angle with the original line(90 degree angle) and that's what you showed in your picture with rotating a point.

Answer 31

It's a bit more complex though, from what I'm learning atm ;P haha

Answer 32

I got y = 5/2x + 23

Answer 33

I simplified yours and got y = -5/2x + 17

Answer 34

Point Slope Form Way:

y - 3 = 5/2(x - -8)

negative rules

y - 3 = 5/2(x + 8)

Property of Distribution

y - 3 = 5/2x + 20

Added 3 to both sides

y = 5/2x + 23

Answer 35

This was from me converting 2x + 5y = -15 into y = -2/5x - 3

Then converting the slope from -2/5x into 5/2x for the negative reciprocal.

Then I did the Point Slope Form Way

Answer 36

@wio

Answer 37

I did the math wrong. \(-5(-8) +2(3) = 40+6=46\)

Answer 38

So I should have gotten:\[
-5x+2y=46\implies y=\frac 52x+23
\]

Answer 39

You were right, just checked my work

Answer 40

I missed something in my equation and I reworked it and I got the same thing as you

Answer 41

My way is a bit different, but it isn't complex.

Answer 42

Basically the trick is \(ax+by\to bx-ay\).

Answer 43

However, some people just like to use formulas and stuff.

Answer 44

\[
y=mx+b\implies y-mx=b\to x+my=b\implies y=-\frac 1m x+b
\]

Answer 45

The whole "negative inverse" \(-1/m\), is actually a result of the other method I gave.

Answer 46

Wio so you mean I was right the whole time!?!?!? ;O

Answer 47

No, if you keep the slope the same it is a parallel line.

Answer 48

I got y = 5/2x + 23, WHEW