Question

Write an equation of the line containing the given point in perpendicular to the given line:
(0,7) 7x+8y=5

Answer 1

put it in standard form Y=mx +b

Answer 2

I still don't get it. y=mx+b just never works out for me, esp with this problem.

The equation of the line is y=_______________what???________ I'm horrible at this, and I'm sorry.

Answer 3

Okay, you have an equation in so-called standard form. You need it in slope-intercept form, which looks like \[y = mx + b\] where \(m\) is the slope and \(b\) is the y-intercept.

\[7x + 8y =5\]Do you know how to solve that for \(y\)?

Answer 4

m = rise/run ...m=y/x

Answer 5

y=0.625???

Answer 6

No, where'd you get that from?

Here's your equation: \[7x + 8y = 5\]You want to rearrange it so that \(y\) is all alone (with no number in front of it), with everything else on the other side of the = sign. Any idea how to start?

Answer 7

apparently not

Answer 8

:(

Answer 9

7x+8y=5 moves things around to make the formula true

Answer 10

y=5/8

Answer 11

Okay, well, you've got a \(7x\) there with the \(8y\), how about moving it to the other side? do you know how to do that?

Answer 12

8y=-7x+5

Answer 13

y=-7/8x+5/8

Answer 14

@creemo do you understand those steps that @rmrjr22 did without explanation? it's okay to say no, I'll explain it if you want.

Answer 15

I do remember that now

Answer 16

I got y by itself.. making thr formula true

Answer 17

you have to stop guessing at numbers, however — you don't have enough information here to find a value for y, only a relationship between y and x.

Answer 18

so, do you need an explanation, or not? we should move on to the next step if you don't, or do the explanation if you do.

Answer 19

which is where finding the perpendicular form comes into play

Answer 20

its important that you can identify slope and the b intercept

Answer 21

Please explain, I am really confused.

Answer 22

okay.

\[7x+8y=5 \]We want \(y\) alone on the left (or the right, it doesn't matter). First step is to move any other terms on the same side with \(y\) to the other side. We can do just about anything to an equation without damaging it, so long as we do the same thing to both sides. Here, we are going to subtract \(7x\) from each side of the equation. Just like a teeter-totter at the park, if add or subtract the same amount from each side, it stays balanced.

\[7x+8y-7x = 5-7x\]\[8y = 5-7x\]Okay so far?

Answer 23

yes

Answer 24

Okay, now we want to get rid of that pesky 8 in front the \(y\). We'll do that by dividing everything by 8. (one of the few things you cannot do, btw, is divide by 0).

\[8y = 5 -7x\]\[8y/8 = 5/8 - 7x/8\]\[y = 5/8 - 7x/8 \]Okay?

Answer 25

Next, I'm going to reorder the terms on the right hand side. This makes no difference to the value, but makes it easier to recognize this as slope-intercept form: \[y = mx + b\]

\[y = 5/8 - 7x/8\]\[y = -7x/8 + 5/8\]

Looking at your equation, and the template for slope-intercept form, can you tell me the values of \(m\) and \(b\)?

Answer 26

it might be a little easier if I reformat:
\[y = -\frac{7}{8} x + \frac{5}{8}\]

Answer 27

m = slope = rise/run =-7x/8

Answer 28

@rmrjr22 no, that's not correct. let's let @creemo answer...

Answer 29

m=-7/8
b=5/8
Is this correct

Answer 30

ding ding ding, we have a winnah! :-)

Answer 31

Okay, so we've determined that the slope of our original line is -7/8. Perpendicular lines have a property that the product of their slopes (*) is always -1. So for our new equation, we need to find a number that, if multiplied by -7/8, gives -1.

(*) If one of the lines is of the form x = <constant> or y = <constant> then this is not true, because a vertical line does not have a slope.

Answer 32

\[-\frac{7}{8} x = -1\]Multiply both sides by 8\[-7x = -8\]divide both sides by -7\[x = -8/-7 = 8/7

Answer 33

for perpendicular lines, dont you just flip the slope? and flip the sign?

Answer 34

that is effectively the same thing, yes, but I think it is important to understand what you are doing, rather than just have a recipe.

Answer 35

@Creemo you still with us?

Answer 36

yes, I'm still trying to write it out and understand

Answer 37

okay. writing it out is good — for some reason, going through the motions of writing can really help.

Answer 38

so, we've determined the slope of our new line, designed to be perpendicular to the old. the new slope is \(m = 8/7\) so our line will look something like \[y = \frac{8}{7}x + b\]Now, we know that the line is to go through the point (0,7). We can find \(b\) by plugging in that point in our new equation and figuring out what value of \(b\) makes it true. That's one way. There's another way which I'll show first, called the point-slope equation. The point-slope equation allows you to directly write the equation of a line with slope \(m\) passing through point \((x_0,y_0)\):
\[y - y_0 = m(x-x_0)\]so in our case, that would be\[y - 7 = \frac{8}{7}(x-0)\]or\[y-7 = \frac{8}{7}x\]and we can tidy it up a bit by adding 7 to each side to give us\[y=\frac{8}{7}x + 7\]
Now we'll test our work: let's make sure \((0,7)\) works in that equation!
\[7 = \frac{8}{7}(0) + 7\]\[7 = 0 + 7\checkmark\]Good, the line goes through the required point.

Now I mentioned that there was another way to do the equation. We had \[y = \frac{8}{7}x+b\]We can plug our known point directly into that and solve for \(b\):
\[7 = \frac{8}{7}(0) + b\]\[7 = 0+b\]\[b = 7\]So our final equation, either way we did it, would be \[y = \frac{8}{7}x+7\]

Answer 39

I do not understand this part:
divide both sides by -7\[x=-8/-7=8/7]

Answer 40

Sorry, I didn't get a character there, that should have been
\[x = -8/-7 = 8/7\]

Answer 41

I actually think I understand all of this now!! I REALLY appreciate you taking all of this time to help!!!!

Answer 42

For your prize, here's a graph of the lines:

Answer 43

You should be able to identify which line is the original and which is the perpendicular, based on your knowledge of their respective slopes. Is the purple line the original, or the perpendicular?

Answer 44

remember, a positive slope means up, up and away... :-)

Answer 45

I believe the purple line is the original. I'm sorry, I didn't realize you replied!!

Answer 46

Wasn't our original line the one with a negative slope? A negative slope means that the line descends as we move to the right along the x-axis.

Answer 47

Think about \[y = x\]That's a line that goes up at a 45 degree angle from the origin, with a slope of 1. Every step to the right you take, you also take one step up. Positive slope.

Answer 48

Also, if you have a line in standard form, such as this one started out\[7x + 8y = 5\]The line crosses the x-axis at the point where y = 0, so \(7x+8(0) = 5\) or \(x = 5/7\), and crosses the y-axis at the corresponding point where x = 0, so \(7(0) + 8y = 5\) or \(y = 5/8\)

If you're observant, you might notice that those spots are C/A and C/B if our equation is in standard form Ax + By = C (which it is). If you're observant but forgetful (like I am) then plugging in the 0 for x and y works as well...

Answer 49

I'm going to print all of this off. You have been the most helpful, and I couldn't be more appreciative!

Answer 50

Great, glad I could eliminate some of the mystery :-)