Question

determine the coordinates of a point on the line with the symmetric equation (x+3)/2=(y-2)/4=(z+4)/3

Answer 1

\[\frac{ x+3 }{ 2 } = \frac{ y-2 }{ 4 } = \frac{ z+4 }{ 3 }\]

Answer 2

These are fun. All you need to do is pick values for x, y, and z that make all sides equal to each other.

Answer 3

We can begin with any of the "fractions" so how about we pick something for x, see what fraction we get, and then try to pick y and z to match.

Answer 4

So, what number would you like to pick for x?

Answer 5

would i need to have common denominators?

Answer 6

Not necessarily, but it isn't a bad idea if you want to go that way.

Answer 7

It would certainly make guessing a lot easier.

Answer 8

Let's try that and see.

Answer 9

ok so \[\frac{ 6x+18 }{ 12 } = \frac{ 3y-6 }{ 12 } = \frac{ 4z+16 }{ 12 }\]

Answer 10

Alright, looking good.

Answer 11

Any ideas on how you want to move forward?

Answer 12

im not too sure. would I combine them to (6x+18+3y-6+4z+16)/12 and then simplify?

Answer 13

Well, we can't combine them quite like that since we don't know exactly what they equal. But we do know that they are all equal to the same thing. Let's call the thing they equal t.

So,
\[\frac{6x + 18}{12} = \frac{3y-6}{12} = \frac{4z + 16}{12} = t\]
Are you ok with that so far?

Answer 14

yes i am

Answer 15

Alright, so what we can do is solve each equation for x, y, and z. I'll do the one for x.
\[\frac{6x + 18}{12} = t\]
so
\[x= (12t -18)/6 = 2t - 3\]

Answer 16

Do you see how I did that?

Answer 17

yes I do

Answer 18

would i do the same for y and z?

Answer 19

Yes do the same for y and z.

Answer 20

y=(12t+6)/3 z=(12-16)/4

Answer 21

I think a t might have escaped for the z, double check that one. :)

Answer 22

Your y is correct though.

Answer 23

Oh I see, I think you meant, z = (12t - 16)/4 right?

Answer 24

oh yes sry about that

Answer 25

So we can simplify our answers and we see that
x = 2t - 3
y = 4t + 2
z = 3t - 4

Does that make sense so far, we are nearly done!

Answer 26

yes i got so far

Answer 27

OK, so now we have the equations to tell us all of the points on the line! Just plug in any t you want and our equations will give you the x,y,z coordinate. For example,

For t = 0,
x = -3
y = 2
z = -4
The point (-3, 2, -4) is on the line.

Answer 28

ok cool! thanks a lot!

Answer 29

Or you could plug in t = 1:
x = -1, y = 6, z = -1 is also a point on the line!

Answer 30

Quick shortcut for you.

Answer 31

We actually didn't need to find common denominators!

Answer 32

Notice that we could just take the original fractions and set them equal to t.

Answer 33

haha just realized!

Answer 34

This gives us the same equations! But I wanted to give you the chance to be in the "driver's seat" and find the solution in your own way. Good work!

Answer 35

thanks!